In a small pan, melt butter over medium heat. Once melted stir in mushrooms and garlic, cook for 5 minutes. Or until mushrooms are softened. Remove from heat, set aside.
Cut panini rolls in half. Lay 2 slices of prosciutto and 1 slice of mozzarella on the bottom half of each roll. Split mushrooms evenly between the two sandwiches. Place other half of roll on top. Brush top and bottom of sandwich with olive oil.
Place sandwiches on panini grill. Close lid. Cook for 5-7 minutes or until bread is browned and cheese is melted.