Brunchburger (two words or one) appears in many sites on google. The Seventh Day Adventists claim its origin, members at the Worthington Cafe, 1969:
Prosage Bunchburgers
Into a large bowl, crumble until mealy, one package of Prosage, then add the following ingredients.
1 large minced green pepper
1 small jar diced pimiento
1 egg
1 4-oz pkg. grated mozarella cheese
dash Paprika
liberal dash garlic salt
1/4 teaspoon oregano
Mix together and form patties.
Fry in vegetable oil until brown. Serve on toasted, well-buttered onion buns.
(Prosage is a vegetarian (ugh!!) substitute for sausage made by Kelloggs).

A lovely combination offered by Dr. Sharp's in Halifax, Canada, served at Brunch (11 AM-2 PM)-
Eggs Benny:
2 poached eggs on toasted English muffin with black forest ham, topped with hollandais sauce and served with home fries.

Unwined in Baltimore offers a brunch burger: Angus beef topped with bacon, cheddar or blue cheese and fried egg, all in a bun, and home fries on the side.

Many other greats; a very rare beef burger, topped with avocado, smoked mayo and argula, called the Origin Burger, at the Origin restaurant in Toronto.