These sandwiches were cooked on a grill, but a skillet would work just as well.
Preheat a grill pan over medium heat. Using a serrated knife, carefully slice ciabatta horizontally into thirds; reserve top and bottom crust for bread slices.
Place bread slices on work surface, crust side down; butter cut side of each slice. Top 1 slice with 6 slices cheese, tomato slices seasoned with salt and pepper, and basil leaves. Top with 6 more slices cheese and then with remaining bread slice, buttered side down. Butter 1 exterior side of sandwich, seasoning with salt and pepper; halve sandwich to fit grill pan.
Place sandwich butter side down onto grill pan. Weight sandwich with a large cast-iron skillet; cook until golden, about 3 minutes. Remove from pan, butter exterior side of top bread slice; turn, weight down sandwich, and cook for 3 minutes more. Repeat with remaining sandwich half. Slice sandwiches crosswise into 2-inch pieces. Serve immediately.