Egg drop soup (also called egg flower soup) is a staple of Chinese American cuisine. The origins of the soup are unclear. Egg drop/flower soup was served in New York’s Chinatown by the early 1900s, but historial citations appear from non-Chinese restaurants as well.
   
“Wonton egg drop soup” adds together wontons (as in wonton soup) and egg drop soup.
 
       
Wikipedia: Egg drop soup
Egg drop soup is best known as a Chinese soup of beaten eggs, chicken broth, and boiled water. Condiments such as table salt, black pepper, and green onion are also commonly added. The soup is finished by adding a thin stream of scrambled egg to the boiling broth, creating thin, silken strands of cooked egg that float in the soup. Egg drop soup using a different recipe is also known as a simple soup in different European countries and Japan.
 
American Chinese cuisine
In the United States egg drop soup is one of the main soups offered in the cuisine. Egg drop soups can be made from scratch. Soup mixes are also commercially available. The cook only needs to add water and beaten egg. The taste is salty.
 
Chinese cuisine
Egg drop soup exist in China as a thinner form compared to its western counterpart. Contrary to popular belief, the dish is relatively rare. It can be mixed with regional ingredients ranging from tofu, scallion or even bean sprouts. 

Mietta’s Recipe Collection
Dahn Far Tong (Egg Flower Soup)
Preamble
“This simple nourishing soup can be made in a few minutes using chicken stock cubes. The beaten egg will set when pour into the boiling soup and look like chrysanthemum petals.” 
 
21 April 1903, Wilkes-Barre (PA) Times, pg. 1:
Hotel Hart.
WEDNESDAY, APRIL 22, 1903.
DINNER.
Soups
English Beef
Consomme Egg Drop
     
18 January 1910, Indianapolis (IN) Star, pg. 14, col. 7 ad:
Egg Drop Soup
(Smith’s—ed.)
 
26 February 1914, Indianapolis (IN) Star, pg. 16, col. 7 ad:
Consomme with Egg Drop…10c
(Budweiser Cafe—ed.)
 
The “Best by Test” Cook Book
compiled by Mrs. Alfred Loeb
for the benefit of
The Hebrew Infant Asylum
New York, NY
1914
Pg. 101:
Egg Drop Soup—Take as much soup stock as desired.  Stir some flour smooth with water, add one or two eggs and a little salt and drop from spoon in the boiling soup just before serving.
     
19 February 1915, Racine (WI) Journal-News, pg. 7, col. 4 ad:
Chicken Egg Drop 5c
(Racine Elgin Lunch—ed.)
 
21 April 1921, Trenton (NJ) Evening News, pg. 2, col. 1 ad:
Consomme Egg Drop
(George’s Restaurant—ed.)
   
19 March 1922, Syracuse (NY) Herald, pg. 17, col. 1 ad:
Consomme with Egg Drop
(Yates—ed.)
   
13 January 1924, Oakland (CA) <>Tribune, “Ways That Are Wary: A Story of Chinese Intrigue” by Samuel De Isra, pg. 4-W, col. 1:
Bo Ch’at, his mind taken up with his plot against Lee Quan, gulped down two small bowls of the strong millet liquor, and followed it with a bowl of hot egg-flower soup, then, leaving his clerks to chatter over their chow mein, their rice cakes and tea, the drug vender went down to the secret room.
 
19 May 1928, Appleton (WI) Post-Crescent, pg. 3, col. 4 ad:
Chicken Egg Drop Soup
(Snider’s—ed.)
 
Tips on Tables:
Being a guide to dining and Wining in New York at 365 restaurants suitable to every mood and every purse
by George Ross
New York, NY: Covici Friede Publishers
1934
Pg. 85: HANG FAR LOW 23 Pell Street (Doyers and Mott Streets)
The table d’hote dinner is 65 cents, and don’t forget to start off with the mushroom and chicken soup with egg drop.
     
Google Books
Chinatown Inside Out
by Leong Gor Yun
New York, NY: Barrows Mussey
1936
Pg. 245:
Tomato soup with eggs (Fan-kia-don-far-tong) 
 
9 December 1938, New York (NY) Herald Tribune, pg. 23, col. 3:
Chinese Dinner Menu.
(...)
Egg Flower Soup.
 
25 January 1940, New York (NY) Herald Tribune, pg. 14, col. 6:
There is egg drop soup and noodle soup thick with shredded chicken.
 
29 January 1940, Portsmouth (NH) Herald, “Man About Manhatan” by George Tucker, pg. 6, col. 2:
NEW YORK—Secrets on Broadway travel faster than favorites at Hialeah. I was sitting in the Beachcombers bar having a bowl of egg flower soup when a man known to all Broadway as Dave came by.
 
27 July 1946, Reno (NV) Evening Gazette, pg. 3, col. 3 ad:
CHINESE DISHES a la carte
SOUP
(...)
DAN FAR TONG egg flower ... 15c
(West Indies—ed.)
14 September 1961, Dallas (TX) Morning News, “‘Egg Flower’ Soup Special,” section 5, pg. 14:
Yellow and white petals of egg, floating on tasty chicken broth, is called “Chrysanthemum” or “Egg Flower” soup by the Chinese. TO make it quickly, experts suggest you bring 3 1/2 cups water to a rolling boil over a high flame.
 
Add a package of dehydrated chicken noodle soup mix and turn the flame back to simmer for five minutes. Mix 1/4 teaspoon cornstarch with 1 tablespoon water and turn the flame to high to bring the soup to a rolling boil.
 
Add the cornstarch paste and a slightly beaten egg. Turn out the flame and serve the soup immediately. Serves four.
   
6 March 1975, Dallas (TX) Morning News, “Egg Flower is Chinese favorite,” section E, pg. 13:
EGG FLOWER SOUP
(Don Fah Tong)
6 cups chicken stock
1/2 cup diced canned mushrooms
1 cup frozen peas
1/2 cup diced uncooked chicken
2 eggs, beaten
Salt
Place chicken stock in a covered pot and bring to a boil. Add peas, mushrooms, chicken. Add eggs and stir until eggs separate in shreds. Season to taste with salt. Serves six.
   
Live Search Books
A Culinary Dictionary: The Chef’s Companion
by Elizabeth Riely
Hoboken, NJ: John Wiley & Sons Publishing
2003
Pg. 88:
dan hua tang [dahn whah wahng] Egg drop soup in Chinese-American restaurants, literally “egg flower soup”; chicken stock into which beaten eggs are whisked to form silky strands that resemble flower petals, garnished with sliced scallions; Western versions often contain more elaborate ingredients.