Whisk the eggs and milk with a fork. Add the cheese to the egg mixture and season with salt and pepper.
Wash the cherry tomatoes and cut in half. Wash the basil too and shake dry. Cut the leaves into small pieces.
Heat a dollop of butter in the pan over medium heat. Using a ladle, pour 1 ladle of the egg mixture into the pan and distribute it.
Scatter four tomato halves and some of the basil on the egg mixture. As soon as the omelette is firm, fold one half over the other with a spatula and carefully lift it out of the pan onto a plate.
Bake more omelettes from the remaining egg mixture.