A “Texas omelet” consists of eggs, onion, chili, tomato, and cheese. The term is somewhat new (with citations mostly starting from the 1970s) and recipes vary.
 
 
Xrecipes.com
Lynn’s Texas Omelet
3 eggs
1 small onion
1 green chilli
1 tomato
salt & pepper to taste
1 tsp milk
1/4 tsp red chilli powder

Heat 1 1/2 Tbs oil in a frying pan and add chopped onions. Fry till golden brown and then add chopped green chili followed by diced tomatoes. Fry till tomatoes extract their juice.

Add red chilli powder and salt & pepper. Beat eggs add salt & pepper , milk and place the beaten eggs into a greased hot frying pan.

When bottom is set place the tomatoes and onion mixture in one half and fold. Cook until both ends are sticked together.
 
Xrecipes.com
Granny’s Texas Omelet
1 (8 inch) flour tortilla
1 med. tomato, peeled and cut up
1 (4 oz.) can green chilies, drained, seeded, rinsed
1/2 sm. onion, cut up
1 tbsp. cilantro
2 dashes cayenne
1 tbsp. olive oil
1 sm. garlic clove
1 sm. avocado, peeled and seeded
1 tbsp. mayonnaise
3 eggs
1 tbsp. water
2 tbsp. butter
1/4 c. shredded Cheddar
Sour cream
 
In a blender, combine tomato, half the chilies, onion, cilantro, cayenne, oil, garlic, 1/4 teaspoon salt, dash of pepper. Blend until smooth. Pour into saucepan and heat to boil. Reduce heat, cook and stir 5 minutes or to slightly thickened. Keep warm.
 
Mash avocado. Stir in mayonnaise, set aside. In bowl, beat together eggs, water, 1/4 teaspoon salt, dash of pepper. Chop remaining chilies and stir into eggs.
 
In 8-inch skillet with flared sides, heat butter. Lift and tilt pan to coat the sides. Add the egg mixture. Cook over medium heat as eggs set, run a spatula around skillet, lifting eggs to allow uncooked portion to flow underneath. When eggs are set, remove from heat.
 
Place tortilla on a plate. Spread with avocado mixture. Sprinkle with 1/4 cup cheese. Invert omelet on top. Roll up tortilla. Top with the remaining cheese and a dollop of sour cream. Garnish with cilantro. Serve with tomato sauce. Makes 2 servings.
     
Texas Bar and Grill (Irving, TX)
Texas Omelet
Our Version of a Denver Omelet. We start with three fresh eggs
And fill them with ham, sauteed Mushrooms, Bell Pepper, Onion
And Mixed Cheese.
6.99
 
Pancake House and Grill (Plano, TX)
Texas Omelet .................. 5.49
(diced ham, green pepper, onion & cheese topped with salsa)
       
Sweet Temptations Cafe (Dallas, TX)
Texas Omelet: $6.99
Green onion, tomato, cilantro, cheddar cheese, green bell pepper, guacamole, salsa, herb potatoes. 
 
San Antonio SAMS: A Texas Grill (Norfolk, VA)
TEXAS OMELET ( $6.99 )
3 eggs with green & red peppers, onions, jalapenos, mushrooms, jack & cheddar cheese, served with a side of bacon or sausage, fiesta potatoes, Texas toast and a fresh fruit kabob.

Texas Monthly (Aug. 2003): Houston Restaurants
Update BENJY’S Benjy’s rates high on our “best brunch” scale. Why? Big portions at modest prices, served in a comfortable, casual modern setting. Witness the Texas omelet hefty with fajita steak, sweet peppers, wild mushrooms, and asadero cheese for $11.95.
   
Country Pride Restaurant, (Kenly, NC)
Twisted Texas Omelet
A new twist on a traditional favorite! This three-egg omelet is filled with grilled chicken, cheddar cheese, roasted vegetables, salsa and sour cream. Served with golden hash browns, Texas Toast and jelly.
       
eBay
EXCELLENT 1946 GAS RANGE COOKBOOK ETHNIC
America Cooks
This is a cookbook from the Hardwick Stove Company, cleveland, Tennessee 1946 I have also included a small pamphlet about the The Hardwick challenger The Rite-Hite Mirror-Magic oven. The cover of the cookbook shows people of all ethnic types. TheCookBook American Cooks has recipes from all over the county from a Texas Omelet to Honey Spiece cake. Inside are more photos of the 1946 ranges. The book has 64 pages size 5 1/2” by 8” it is in excellent never used condition.
 
23 March 1975, Mansfield (Ohio) News Journal, pg. 2B:
The Kempton Texas Omelet ingredients are whipped eggs, green peppers, onions and cubes of ham blended and fried on a griddle.
 
3 April 1982, Frederick (MD) Post, “The Extension Line” by Kathryn Jenkins, pg. B4:
Texas Omelet for Three
6 eggs
8 tablespoons milk—salt and pepper
1 tablespoon green pepper
1 tablespoon minced onion or endives
2 tablespoons grated cheese
2 slices fried bacon crumbled
2 tablespoons mushrooms
(You may use any or all of the ingredients or any others that suit your taste.)
Beat eggs until slightly fluffy. Add milk and a dash of salt and pepper. Add other ingredients. Pour into a 300 degree skillet. Cook slowly. As under surface becomes set, hit eggs with spatula so uncooked portion goes underneath and cooks. As soon as texture is set, fold over and serve.