First, assemble the sandwiches: On each of 4 bread slices, place 3 slices of cheese. Season with salt, then top with 2 tomato halves, 4 basil leaves, and another slice of bread.

Heat 2 large nonstick skillets over moderately high heat. When hot, add 1 tablespoon of the butter to each skillet and swirl to melt the butter and coat the skillet. Put 2 sandwiches in each skillet and place a heavy skillet on top as a weight; if the upper skillet isn't heavy, put more weight, such as canned goods, inside it. Cook until the sandwiches are well browned on the bottom, 2 to 3 minutes, then remove them. Add another 1/2 tablespoon butter to each skillet, and return the sandwiches, browned side up, to the skillets. Weight as before and cook until the second side is well browned, about 3 minutes longer.

Cut the sandwiches in half on the diagonal and serve immediately.