Spread 1 teaspoon quince paste on each of 8 pieces of bread. Divide the cheese slices among the spread pieces of bread, then top with the remaining portions of bread to form 8 small sandwiches. Heat 1 tablespoon of the butter in a large skillet and, when melted add half of the sandwiches. Gently press down with a spatula once or twice during the cooking. When one side is golden, add the remaining tablespoon of butter, turn the sandwiches over, and cook until crisp and cheese is melted through. Slice the sandwiches in half or in quarters and serve immediately. Repeat with remaining butter and sandwiches.