Mince 2 of the garlic cloves. Combine minced garlic, Champagne vinegar, 1/2 cup of the olive oil, 1 1/2 teaspoons of the salt, and 1/2 teaspoon of the pepper in a large ziplock plastic freezer bag. Add cherry tomatoes, green tomatoes, and mozzarella. Seal bag, and shake gently to coat. Chill 1 hour, turning occasionally.
Combine basil, balsamic vinegar, shallot, mayonnaise, sour cream, 6 tablespoons of the olive oil, and remaining 3/4 teaspoon salt and 1/4 teaspoon pepper in a blender. Process until smooth, about 45 seconds.
Preheat grill to medium-high (400°F to 450°F). Remove tomatoes from marinade, reserving marinade. Grill tomatoes until tender and grill marks appear, about 2 minutes per side. Set aside.
Brush bread slices with remaining 2 tablespoons olive oil. Grill until grill marks appear, about 1 minute per side. Rub each toasted bread slice with remaining garlic clove.
Place grilled bread on a serving platter. Arrange grilled tomatoes and marinated mozzarella over bread. Drizzle with reserved marinade and basil dressing. Sprinkle with basil leaves.