In a small bowl whisk together eggs and milk, season with salt and pepper. Heat butter in a large non-stick skillet over med-high heat. Pour in egg mixture slowly. Eggs should begin to set immediately.
Using a rubber spatula push cooked portions from edges into center of pan. Continue around the pan so the uncooked eggs are exposed to hot surface. Tilt pan as needed while exposing uncooked portions.
Once the eggs are almost cooked through gently spread the pesto on one side of the omelet. Top with tomatoes and shaved Parmigiano Reggiano.
Fold the omelet in half and slip onto a plate. Repeat for the second omelet.
Top with more shaved Parmigiano Reggiano.