Grilled cheese sandwiches are almost never grilled. They're usually pan-cooked in butter while weighed down with some kind of heavy object (or pressed down with a spatula).
Other options include the George Foreman grill, which, while not much good for approximating outdoor grilling, is pretty good at melting cheese between slices of bread, and the one-trick-pony panini press. Creative cooks may compress and heat cheese sandwiches in a waffle iron, or, a la Benny and Joon, a clothes iron.
What all of these methods have in common is the assumption of a closed-face sandwich: a filling enveloped by two soft slices of bread. Granted, most conventional sandwiches are based on this assumption -- but it becomes problematic when you get into grilled cheese territory. That's because you can put only so much cheese in a grilled cheese sandwich before it starts spilling out the sides and burning on whatever surface you've chosen for heating it.
The obvious solution is to make your grilled cheese sandwiches open-faced and broiled, on a foundation sturdy enough to support as much cheese and as many additional ingredients as your heart desires. For the bread, I like a ciabatta roll, which though not exactly traditional, is hefty and crusty enough that almost no amount of moist toppings will render it floppy. (Other excellent choices: a section of a baguette or a square of thick focaccia.)
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The best cheese to use in grilled cheese sandwiches is a matter of everlasting controversy, but there is a very strong case to be made for taleggio. You can think of this soft-rinded Italian masterpiece as the Steely Dan of cheese: seriously funky, yet smooth and accessible. Unlike cheeses that lose their sharpness (like cheddar) or wonderfully dense texture (like gorgonzola) when melted, taleggio gets even creamier, richer and tangier when broiled.
As for accouterments, sauteed mushrooms are hard to beat; they contribute both savory flavor and robust texture, yet they still complement, rather than compete with, the cheese. They also take well to other meaty seasonings, so if you'd rather saute them in bacon or pancetta fat instead of olive oil, be my guest.
Grilled taleggio sandwich with sauteed mushrooms
1 tablespoon extra-virgin olive oil
1 small red onion or large shallot, chopped
1 garlic clove, minced
Salt and black pepper
4 ounces fresh mushrooms, preferably a mixture, trimmed and roughly chopped
1 teaspoon chopped fresh rosemary or 1/2 teaspoon crumbled dried rosemary
1 large ciabatta sandwich roll, split horizontally
3 ounces taleggio cheese, thinly sliced
1. Heat the oven (or a toaster oven) to 450 degrees. Put the oil in a large skillet over medium-high heat. When it's hot, add the onion and garlic and season with salt and pepper. Cook, stirring occasionally, until the onion is soft and golden, about 10 minutes. Add the mushrooms and rosemary and continue to cook, stirring occasionally, until all the mushrooms' liquid has evaporated, 8 to 10 minutes.
2. Spread the mushroom mixture evenly over the two halves of the ciabatta roll, top with the taleggio, and transfer to a baking sheet. Bake until the taleggio is melted and lightly browned, about 5 minutes. Serve immediately.
Nutritional information per serving, based on 1: 684 calories, 38 grams fat, 59 grams carbohydrates, 30 grams protein, 76 milligrams cholesterol, 743 milligrams sodium, 6 grams dietary fiber, 49 percent of calories from fat.