A mashed potatoes riddle is:
 
Q: Why did the farmer plow his field with a steamroller?
A: He wanted mashed potatoes.
 
The joke has been cited in print since at least 1968.
 
         
Google Books
Jokes, Puns, and Riddles
By David Allen Clark
Illustrations by Lionel Kalish
Garden City, NY: Doubleday
1968
Pg. 44:
City Slicker: Mr. Farmer, why are you plowing your field with a steam roller?
Farmer: I’m raising mashed potatoes this year.
   
18 July 1971, Boston (MA) Herald Traveler, Young Folks Page, pg. ?, col. 7:
City Slicker: Why are you plowing your field with a steam roller?
Farmer: I’m raising mashed potatoes this year.
   
18 June 1973, The Palladium-Item (Richmond, IN), “Mini Jokes,” The Mini Page, pg. 4, col. 1:
Q. Why are you plowing your field with a steam roller?
A. I’m raising mashed potatoes this year!
Martha Helderman, Richmond.
 
Google Books
What’s to Eat?:
And Other Questions Kids Ask about Food
U.S. Department of Agriculture
1979
Pg. 53:
Sue: Why did the farmer go over his field with a steamroller?
Lou. He wanted to raise mashed potatoes.
     
9 April 1980. Pittsburgh (PA) Press “‘What;s to Eat?’ Answers Many Questions” by Marilyn McDevitt Rubin, pg. D-1, col. 3:
Her: Why did the farmer go over his field with a steamroller?
Him. He wanted to raise mashed potatoes.
 
10 April 1980, Lexington (KY) Herald, “Just kidding,” pg. A-26, col. 5:
Why did the farmer go over his field with a steamroller? He wanted to raise mashed potatoes!
 
Google Books
1001 More Cool Jokes with Glen Singleton
By Barb Whiter
Victoria, Australia: Hinkler Book Distributors
2002
Pg. 109:
Why did the farmer plow his field with a steamroller?
He wanted to grow mashed potatoes.
 
4 May 2002, Evening Mail (Birmingham, UK), “Children: Funny Side,” pg. 32:
Why did the farmer plough his field with a steamroller? Because he wanted to grow mashed potatoes!
   
Google Books
Super Goofy Jokes
By Jacqueline Horsfall
Illustrated by Rob Collinet
New York, NY: Sterling Publishing Company, Inc.
2004
Pg. 26:
Why did the farmer plow his field with a steamroller?
He wanted to grow mashed potatoes.
 
Twitter
Whitney Segura
@gardeningfun
Q: Why did the farmer plough his field with a steamroller?
A: Because he wanted mashed potatoes.
7:25 PM - 24 Apr 2009
   
Twitter
Bollywood Sargam
@bollywoodsargam
Farmer jokes - 4806 Joke: Did you hear about the farmer you ploughed his field with a steamroller ? He wanted .. http://bit.ly/1EoutQ
12:20 PM - 30 Aug 2009
     
YouTube
Jokes - Why did the farmer plow his field with a steamroller? He wanted mashed potatoes.
jokesus
Published on Oct 2, 2015
Joke :
Why did the farmer plow his field with a steamroller? 
He wanted mashed potatoes.
 
YouTube
Ok, here is the joke. Did you hear about the farmer you plough…
botcaster inc. bot
Published on Jan 22, 2016
Did you hear about the farmer you ploughed his field with a steamroller ?
He wanted to grow mash potatoes!
 
Twitter
lorraine Jo King
@LahLahBean
#sillysaturday
Did you hear about the farmer you ploughed his field with a steamroller?
He wanted to grow mash potatoes!
11:25 PM - 22 Jun 2018