1. In a 10- to 12-inch nonstick frying pan over medium-high heat, stir sausage, onion, and chilies until meat is crumbled and browned, about 15 minutes. Drain off and discard fat.
2. In a bowl, beat eggs to blend with 1/3 cup water, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Pour eggs into meat mixture in frying pan; with a wide spatula, stir over medium-high heat, scraping pan bottom often, until eggs are set to your taste, about 3 minutes for creamy eggs.
3. Lay flour tortillas flat on a counter. Spoon an equal portion of the meat and egg mixture in a band down the center of each tortilla to within 1 inch of opposite edges. Sprinkle mixture evenly with cheese. To form each burrito, fold the bare 1-inch-wide tortilla edges over filling, then roll snugly from an open edge to enclose. Set burritos slightly apart, seams down, in a 10- by 15-inch pan; seal pan with foil. If making up to 1 day ahead, chill.
4. Bake in a 200° oven until cheese is melted (cut a slit to test), 15 to 20 minutes; if chilled, bake in a 325° oven until hot in the center, about 15 minutes.
5. Set burritos on plates and ladle about 1/2 cup hot chili con queso salsa on and around each. Season to taste with salt and pepper.