A “Boston Shake” (or “Boston Milkshake”) was on the menu at Tastee-Freez (a soft serve franchised chain) by at least 1954. It was described in the New Castle (PA) News on April 20, 1956:
 
“FOR BOSTON SHAKES
The Triple Thick Milk Shake
With A Sundae On The Top!”
 
It’s not certain if Tastee-Freez was the first to serve a Boston Shake, but it definitely popularized the item. The reason for the name “Boston” is not known; perhaps the inventor was thinking of a Boston frappe.
 
Fanci Freez opened in Boise, Idaho, in 1947, and has used “Home of Boston Shakes” on its sign. It’s not known, however, when the Boston Shake was first added to its menu.
 
The Snowman (an ice cream shack in Troy, New York) began in the 1950s, and has on its menu a Boston Shake. The Food Network—The United Plates of America: New York considers this a regional dish, but the Boston Shake has clearly been served in other states. There is no evidence that the Snowman served the first Boston Shake.
 
         
Wikipedia: Tastee-Freez
Tastee-Freez is a soft serve franchised chain of 23 fast-food restaurants. Its corporate headquarters is in Newport Beach, California, and it has stores in 12 of the United States, with most of its freestanding stores located in Virginia, Illinois and Maryland. The first Tastee-Freez was established in Keithsburg, Illinois.
 
History
Tastee-Freez was founded in 1950 in Joliet, Illinois, by Leo S. Moranz and Harry Axene.
     
Food Network—The United Plates of America: New York
Boston Shake
Not just a shake and little to do with Beantown, the Boston Shake is one of those hybrid dishes that long predates the hip hybrid-food trend. It’s a specialty at the Troy summertime favorite, the Snowman. Since the early 1950s, the seasonal ice cream stand has made incredible hard-packed homemade ice cream in a variety of flavors, from classic vanilla and mint chip, to Blue Moon and salty caramel. The Boston Shake combines a traditional milkshake and a hot fudge sundae. Each one starts with any flavor shake, layered with a scoop of ice cream, rich hot fudge, airy whipped cream and a cherry on top.
   
Fanci Freez (Idaho)
Boston Shake
shake with a sundae on top! choice of chocolate or caramel
XS 3.39 | S 4.19 | M 5.09 | L 6.19 | XL 7.29
 
Valley Dairy Freeze (Louisville, KY)
What is a Boston Shake?
A Boston Shake is a unique item to Valley Dairy Freeze. It is a combination of shake/malt and a sundae, in either a 14 (small) or 20 (large) oz sizes. You get to create your very own and unique combination and can be as crazy and wild as you want with the choices.
   
27 March 1954, Elwood (IN) Call-Leader, pg. 8, col. 7 ad:
Brown Derby’s, Malts and Shakes, Tastee Freez Ice Cream Sandwiches, all flavors of Boston Shakes, Quarts, Pints and Gallons.
(Tastee-Freez.—ed.)
 
30 April 1954, Coschocton (OH) Tribune, pg. 16, col. 7 ad:
Or ask one of the boys to make you a Strawberry Boston Shake. This is one of our more popular Milk Shakes made with a generous serving of Taste-Freez floating on top. Topped with Berries…fresh frozen, of course.
35c
(Wayne’s Tastee-Freez.—ed.)
 
20 April 1956, New Castle (PA) News, pg. 12, col. 6 ad:
tastee-freez
FOR BOSTON SHAKES
The Triple Thick Milk Shake
With A Sundae On The Top!
 
6 June 1956, Linton (IN) Daily Citizen, pg. 3, col. 1 ad:
BOSTON SHAKES
BANANA SPLITS
TASTEE-FREEZ
HI-WAY 54 EAST
 
Google Books
23 July 1956, Life magazine, pg. 16 ad:
tastee-freez
BOSTON SHAKE
 
18 September 1964, The Daily Inter Lake (Kalispell, MT), pg. 6, col. 2 ad:
Each are Tastee Freez guests for a BOSTON SHAKE or a BANANA SPLIT.
   
5 August 1985, Northwest Herald (Crystal Lake, IL), pg. 7, col. 4 ad:
English Toffee
Boston Shake
Turtle Sundae
(The Freeze.—ed.)
       
28 August 1993, Ithaca (NY) Journal, “Toad’s is hoppin’ in Freeville” by David Hill, pg. 8A, col. 3:
When he was planning Toad’s restaurant, Dave Perkins say he wanted to resurrect ice cream treats from when he was growing up. Some came about from customer suggestions. None will make you slim. Here are a few:
(...)
Boston milkshake: A milkshake with a sundae on top.
 
Roadfood Forums
Boston Shakes
Tristan Indiana 
Sun, 02/8/04 4:11 PM
Is anyone aware of something called a boston shake? Its a milkshake with a sundae on top of it. When I was little, an ice cream stand use to make them but I thought it was something they had made up because I have never seen them anywhere else. But I was at the Triple X Root Beer stand next to Purdue yesterday and they had them on the menu. Does anyone know where else they are served? I lived in Boston for a year but never noticed them there.
 
All Over Albany
The Boston Shake at The Snowman
posted Aug 9, 2011
By Daniel B.
(...)
But The Snowman’s Boston Shake has nothing to do with Boston. In fact those in eastern Massachusetts I spoke to about this concoction were completely perplexed. So what is it? Well, there is a sign predominantly displayed above the ordering window that describes it as:
 
Any flavor milkshake with a scoop of our homemade ice cream, some tasty hot fudge, delicious whipped cream, and a cherry to top it off!
 
For those who need it spelled out for them, that would be a milkshake crowned with a hot fudge sundae. Oh, yes.
     
Idaho Statesman (Boise, ID)
This iconic Boise burger joint will open a second location in — where else? — Meridian
BY DANA OLAND
.(JavaScript must be enabled to view this email address)
October 19, 2017 12:51 PM
Updated October 20, 2017 04:59 PM
Fanci Freez was the first of Boise’s burger and milkshake drive-ins to open back in the day. From 1947 until today, it has been owned by several people. Its reputation fluctuated over the years, going from dive to darling to vintage treasure today.
(...)
Fanci Freez is known for classic “giant” burgers and cheeseburgers, soft-serve ice cream cones and milkshakes in 50 flavors, including the “Boston shake” — a milkshake and a sundae in a single cup.
 
518 Bites & Sights
05/09/2018
FOODS UNIQUE TO UPSTATE NEW YORK & WHERE TO GET THEM
(...)
Boston Shake, The Snowman: You’ve probably gotten ice cream from The Snowman in Troy before but have you ever tried the Boston Shake? Imagine it, two delicious things: a sundae and a shake put together.