Brush peaches with 2 teaspoons olive oil. Grill over medium-high heat until flesh is charred. Remove, let cool slightly and then cut into wedges.
Arrange peaches, tomatoes and mozzarella on a large platter. Add prosciutto and a few basil leaves.
Combine 2 cups basil leaves, garlic, lemon juice, salt, pepper in a blender or food processor and pulse to combine. Add olive oil and blend until everything comes together. Taste and adjust salt as needed - you want this to be super punchy and flavorful!
Drizzle platter with basil vinaigrette, serving the rest on the side.