Pre-heat the oven to 400ºF. Season the chicken with salt and pepper.
Heat 1 tablespoon olive oil over medium to medium-high heat. Add the butter and garlic clove and melt the butter into the oil. Add the chicken and lightly brown, 2-3 minutes per side, then remove to a plate.
Sprinkle the chopped thyme over the chicken and top each piece with a slice of cheese, then wrap with the speck or prosciutto di Parma. Arrange the chicken on a baking sheet and drizzle with about 1 tablespoon olive oil to lightly coat. Bake for 15-20 minutes and remove; switch on the broiler.
Arrange the tomatoes on a baking sheet and cover them with the Parmigiano Reggiano cheese. Broil until golden, 3 minutes or so. Arrange two slices of broiled tomatoes on each chicken breast, garnish with torn basil and serve.