Prepare barbecue (medium-high heat).
Whisk extra virgin olive oil, balsamic vinegar, and pressed garlic clove in small bowl. Season dressing to taste with salt and pepper.
Arrange portobello mushrooms on rimmed baking sheet, gill side up. Sprinkle with salt and pepper. Spoon generous tablespoon dressing into each mushroom; swirl to coat.
Stir ricotta cheese, 1/2 cup parmesan cheese, and 2 Tb chopped fresh basil in small bowl to blend. Season to taste with salt and pepper.
Arrange tomato slices on plate; sprinkle with salt and pepper.
Place mushrooms on grill, gill side down. Grill until edges begin to soften, 3-5 minutes, depending on thickness of mushrooms.
Transfer mushrooms to same baking sheet, gill side up. Nestle one tomato slice into each mushroom.
Divide ricotta cheese mixture atop sliced tomatoes, spreading to cover, about 3 Tb per mushroom. Top with Fontina cheese slices, dividing equally. Sprinkle 1 Tb parmesan cheese over each.
Carefully return mushrooms to grill. Cover barbecue and cook until mushrooms are soft and cheese is melted, about 5 minutes. Sprinkle remaining chopped fresh basil over.
Place mushrooms on plates. Drizzle remaining dressing around mushrooms and serve.