Preheat the oven to 350°F.
In a small bowl, stir together the butter, garlic, fresh basil, pepper flakes, and salt.
Generously spread the garlic-butter mixture on one side of each slice of bread. Arrange the slices of bread on a sheet pan, garlic butter–side up, and bake until golden and fragrant, about 10 minutes. Remove the pan from the oven.
Top the garlic-butter side of each of six slices of bread with two slices mild Cheddar and a generous portion of extra-sharp Cheddar. Close the sandwich with the other slices of bread, buttered-side down. Spread a thin layer of mayo on both unbuttered sides of the sandwiches.
Heat a medium skillet over low heat. Lightly grease the pan with a small amount of butter or cooking spray. Working in batches, add the sandwiches to the pan. Cover the pan with a lid and toast until the first side is golden brown, about 3 minutes. Flip the sandwiches, cover the pan, and toast the other side until golden and the cheese is thoroughly melted, about 3 minutes more. Keep a close eye on things—the sandwich can burn, even on low heat!
Cut the sandwiches as desired, ladle the tomato soup into bowls, drizzle with some heavy cream, and garnish with fresh basil leaves. Serve with the grilled cheese on the side—dipping in the soup is encouraged!