Toss the cubed avocado, tomato, and cheese together in a small bowl. Season with salt and pepper to taste.
In a second bowl, lightly beat the eggs with the chives.
Melt the butter in a skillet over medium heat. Add eggs. Cook without stirring until the omelette starts to bubble around the edges. Gather the mixture toward the center of the pan. When the bottom is set, gently add avocado mixture over half the omelette. Fold the remaining half over the avocado and cook until the filling is heated through. Slide the omelette onto a plate, and serve.