Preheat oven to 350°F
Place chicken breasts between two layers of wax paper and pound with a meat mallet to ¼-thick cutlets.
Heat 2 teaspoons of the olive oil in a small skillet over medium-high heat. Add the mushrooms and sauté until lightly browned, 2 to 3 minutes. Season with salt and pepper, to taste; let cool. Arrange one slice of prosciutto over each cutlet. Sprinkle fontina cheese over prosciutto, dividing it equally.
Mound one quarter of sautéed mushrooms in the center of each cutlet. Fold the long sides of the cutlet toward the center, then roll up, enclosing the mushroom filling. Tie the chicken rolls with kitchen twine and place them seam side down on a small baking pan that has been lightly oiled. Rub the chicken roll with 2 teaspoons of the olive oil and season with salt and black pepper to taste.
Bake for 25 minutes or until chicken is cooked through. (Internal temperature should register 165 F when measured with a meat thermometer.)
Meanwhile, heat the remaining 2 teaspoons of olive oil in a small skillet over medium-high heat. Add the garlic and sauté until golden. Add the tomatoes and cook for 2 minutes. Stir in the basil and season with salt and black pepper to taste.
When the chicken is ready, remove the twine and slice each roll into ½ inch thick slices. Fan the slices out on four individual dinner plates and serve each with a spoonful of the tomato-garlic sauce.