Preheat and Prepare Mushrooms: Preheat oven to grill/broil on high heat. Position an oven shelf in the middle of the oven.
Make Garlic Butter: In a small saucepan or microwave-safe bowl, combine butter, crushed garlic, and chopped parsley. Melt until garlic is fragrant. Brush the bottoms of each mushroom cap with garlic butter and place them, buttered side down, on a baking sheet.
Flip the mushrooms and brush any remaining garlic butter over the insides of each cap. Fill each mushroom with mozzarella slices and tomato slices.
Bake: Broil for about 8 minutes, or until the cheese has melted and is golden.
Make Balsamic Glaze (if homemade): While mushrooms are baking, combine balsamic vinegar and brown sugar (if using) in a small saucepan. Bring to a boil over high heat, then reduce to low and simmer for 5-8 minutes, or until thickened. If not using sugar, simmer for 12-15 minutes.
Serve: Top each mushroom with shredded basil, drizzle with balsamic glaze, and sprinkle with salt to taste.