This perfectly self-contained wrap sandwich can go anywhere. To make a meat-filled version, substitute four ounces of your favorite cold cut for the omelet.
Lightly whisk together eggs, salt, and pepper in a small bowl. Heat butter in a 12-inch nonstick skillet over medium heat. Add beaten eggs; cook, lifting up cooked egg with a spatula to let raw egg stream underneath. Cook until set, about 5 minutes. (If top is still runny, heat omelet in a 350 degree oven until set, about 5 minutes.)
Cut omelet into four strips. Spread each wrap with 1 tablespoon mayonnaise; divide omelet, avocado, carrot, and basil among wraps. Roll up wraps, folding in sides. Halve wraps; wrap in waxed paper. Refrigerate until ready to serve.