Heat butter in a 10-inch nonstick skillet over medium heat.
Whisk eggs, and season with a pinch of salt and pepper. Pour into pan; cook, whisking constantly and shaking pan vigorously until eggs begin to set but are still quite loose. Stop whisking; continue to cook until almost completely set, 1 to 1 1/2 minutes, depending on desired consistency. Sprinkle cheese over eggs; spoon salsa down center of eggs. Reduce heat; let stand until cheese is completely melted. Scatter cilantro over filling.
Slide omelet out of pan, centering it over tortilla. Roll up tortilla, burrito style, and serve with avocado slices on the side.