Spread the butter on one side of each bread slice.
Heat a large skillet over medium heat. Put 2 bread slices in the skillet, buttered-side down.
Layer 2 slices of prosciutto, 1 slice of cheese, one-quarter each of the dates and basil, and another slice of cheese on each piece of bread. Close each with another piece of bread, buttered-side up.
Cook, pressing the sandwiches occasionally with a spatula, until the cheese melts and the bread is golden, 3 to 4 minutes per side. Repeat to make 2 more sandwiches.
