Heat 2 tablespoons of the butter in a shallow 10-inch nonstick frying pan. Add the potatoes to the pan and cook, turning occasionally, until golden brown and tender; remove from pan. Heat the remaining butter in the pan, add the onion and prosciutto cook, stirring, until the onion is soft. Return the potatoes to the pan, mix well and then press the mixture flat in the pan. Top with the tomatoes and chicken.
Combine the eggs, salt and pepper, cheeses and basil and mix well. Gently pour the egg mixture over the chicken mixture in the pan. Cook over a low heat until the base is golden brown (about 10 minutes). Place the pan under a hot broiler (grill) to set and brown the top - cover the pan handle with foil if it is plastic.
Loosen the edges of the frittata, turn onto plate or board and cut into wedges to serve.
This recipe can be made up to a day ahead.