In a small bowl, whisk together mayonnaise, horseradish, mustard, and pepper. Spread mayonnaise mixture onto both sides of each bread slice.
In a large skillet, melt 1 tablespoon butter over medium heat. Reduce heat to medium-low. Place 2 bread slices in skillet. Arrange Brie in a single layer on one bread slice. Arrange a few slices of mozzarella in a single layer on other bread slice. Top one bread slice with 4 persimmon slices, overlapping if necessary. Top with 4 basil leaves. Cover with lid, and cook until cheese is melted, about 1 minute.
Using a spatula, turn one bread slice onto other bread slice, cheese side down. Increase heat to medium. Cook for 30 seconds; turn sandwich over, and cook 30 seconds more. Repeat with remaining ingredients. Serve warm.
Recipe by Taste of the South at https://tasteofthesouthmagazine.com/persimmon-grilled-cheese-sandwiches/