Prepare the Balsamic Dipping Sauce: Heat vinegar in a saucepan over medium heat for 5-8 minutes, until reduced by at least half. Remove from heat, add butter and stir until melted. Transfer to small bowl and set aside until ready to serve.
Preheat oven to a low temperature (around 175 degrees F). Place a cooling rack on a sheet pan. Set aside. (This will be used to keep the sandwiches warm in between batches. If only making a few sandwiches, skip this step.)
Lay out 6 slices of potato bread on work surface. Top each slice with one piece (or two ounces, shredded) of cheese. Top with remaining bread slices. (You will have two sandwiches of each cheese variety.) Brush the top of each sandwich with melted butter.
Preheat griddle or pan over medium heat. Working in batches if necessary, place sandwiches buttered side down onto hot surface. Brush tops of sandwiches with remaining butter. Cook 2-3 minutes per side or until golden and cheese is melted. (If working in batches, place cooked sandwiches on the prepared sheet pan and keep warm in oven until all sandwiches are cooked.)
Once all sandwiches are cooked, remove sheet pan from oven and allow sandwiches to cool enough to handle. Slice each sandwich into three strips and serve with various dipping sauces.
— White American Grilled Cheese with Balsamic Dipping Sauce
— Taco Cheddar Jack Grilled Cheese with Ranch Dressing
— Mozzarella Grilled Cheese with Marinara Sauce
Feel free to experiment with other varieties of cheese and sauces. Other ideas include: Sharp Cheddar + BBQ Sauce, Classic American + Tomato Soup/Puree, Pepper Jack Cheese + Apricot Jam.