A “cook-off” is a cooking competition. Cook-offs often limit themselves to a particular food, such as “chili cook-offs.” The term “cook-off” began in 1936 at the fifteenth Women’s Exposition of Arts and Industry, held at New York City’s Grand Central Palace. 
   
The term “cook-off” perhaps inspired Pillsbury to call its baking competition (started in 1949) a “bake-off.”
 
   
Wikipedia: Cook-Off
A Cook-off is a cooking competition where the contestants each prepare dishes for judging either by a select group of judges or by the general public. Cook-offs are very popular among competitors (such as restaurants) with very similar dishes and serves as a way to decide which recipe is the best for that particular dish.
 
Chili cook-off
A chili cookoff is a social event, similar to a barbecue, in which competitors prepare their own particular recipe for chili con carne and submit it for taste testing. A cookoff may be an informal gathering with the simple goal of sharing recipes and enjoying food, or it may be a large-scale event with an official panel of judges and substantial prizes for winners.
 
The Chili Appreciation Society International (CASI) sanctions over 550 cook-offs annually that raise over $1,000,000.00 for charity. Corporate CASI, which has the function of sanctioning all of these cook-offs, doing the necessary work to tabulate cookoff results, and putting on the annual Terlingua International Chili Championship, has an annual income of approximately $250,000.00. Out of this annual income around $60,000.00 goes to charity. The CASI is a 501(c)(3) corporation.
 
The International Chili Society (ICS) organizes chili cook-offs as fundraisers. Its annual cook-offs are on a grand scale, with regional qualifying events that attract tens of thousands of participants and a “World Championship” cook-off held in October. The ICS has raised millions of dollars for charity while still awarding hundreds of thousands of dollars to winners.
 
As early as 1970, many communities have hosted non-ICS cook-offs, leading to a competing “World Championship” cook-off in Terlingua, Texas.
 
Notable cook-offs include:

. the Chilympiad in San Marcos, Texas (for men only)
. the Hell Hath No Fury Like A Woman Scorned cookoff in Luckenbach, Texas
. the Chili Appreciation Society International Chili Championship in Terlingua, Texas
. the Frank X. Tolbert/Wick Fowler World Chili Championships in Terlingua, Texas
     
Merriam-Webster Dictionary
Main Entry: cook–off
Pronunciation: \ˈku̇k-ˌȯf, -ˌäf\
Function: noun
Date: 1937
: a cooking competition  
 
(Oxford English Dictionary)
cook-off, n.
orig. U.S.
[< COOK v.1 + OFF adv., after PLAY-OFF n. Cf. BAKE-OFF n.]

A cookery competition, esp. one between non-professional contestants.
1936 Port Arthur (Texas) News 3 May D4/3 There will be a grand cook-off with five sectional winners competing.
1973 News (Mexico City) 29 July (Vistas Suppl.) 5/4 Included in the event are a cookoff, taste-in (for $1 one can sample the top five recipes in each of the three categories), country dinner ($2.50 for roast beef, coleslaw, cornbread and black-eyed peas), gospel singing.
1990 Good Housek. May 33/2 Three finalists were invited to GH for the cook-off, which was judged by..[a] restaurant critic.
2001 Mail & Guardian (Johannesburg) 18-24 May (Friday Suppl.) 2/2 The Good Food and Wine Show..promises to be the hub of action, hosting a battery of exhibitions, chef master-classes, cook-offs and special foodie events.
 
24 May 1936, New York (NY) Times, “‘Cook-Off’ Contest Here; Culinary Experts From Various Sections of Nation to Vie at Grand Central Palace,” pg. N6:
(The fifteenth Women’s Exposition of Arts and Industry at the Grand Central Palace—ed.)
Cooking Contest Tuesday
Tuesday morning will bring one of the primary features of the week, the “cook-off,” in which culinary artists from each of half a dozen sections of the country will demonstrate their finest recipes for canned, time and labor saving foods, in the finals of the annual cooking contest. A jury of experts seated at a dinner table in a booth near by the model kitchen where they will work, on the main floor, will receive the dishes as they come from the ovens, for tasting and rating as to flavor, color and texture.
 
27 May 1936, New York (NY) Times, “Pot-Roast Dinner Wins Kitchen Title, pg. 25:
Challenging the appeal of such dishes as the “Chicken Symphony” and conquering even the erstwhile supreme broiled sirloin steak with mushrooms, a good old-fashioned main course of pot roast, string beans, egg noodles and escalloped tomatoes won first price last night in the “cook-off” of the women’s national menu recipe competition in Grand Central Palace.
 
29 March 1937, Christian Science Monitor, “Women’s ‘Cook-Off’ Put on Air,” pg. 1:
NEW YORK, March 29—Six housewlves from different parts of the country will face a battery of newsreel cameras, microphones and observers here this week for the title of national cooking champion of 1937. Every skirl of the egg beater, every dredging of flour, every…
   
4 August 1939, Christian Science Monitor, “Four Reach Finals in Egg ‘Cook-Off’,” pg. 4:
CLEVELAND, Ohio, Aug. 4 (AP)—Mrs. Homer Hixson of Gainesville, Fla., received the highest rating yesterday among six entrants…