Not White, Wheat, Or Sourdough — Grilled Cheese Is 10x More Flavorful When You Use This Bread
In most grilled cheese sandwiches, the cheese provides the majority of the flavor, with the bread (especially if it's plain and white) adding some crunch but not much more. Instead, consider using a more robust, punchier loaf: rye bread.
Darker than white or wheat flours, rye flour gives bread a nutty, earthy note, along with a slight maltiness. It can be a little tangy, with a sourness that can balance out the cheese's rich fattiness. It's usually denser than your average supermarket bread and will add some bulk to your grilled cheese (so if you really prefer your bread light and fluffy, rye may not be for you). Some people think rye has a licorice flavor, but that comes from the caraway seeds, which are often added to rye loaves but aren't necessarily a key component.
Bear in mind that there's light and dark rye out there; go for light rye since the dark bread holds more moisture, making it less ideal for grilling. But that's just the bread side of the equation: Since rye has a distinct flavor, you'll also want to think about what cheese(s) to put inside.
Some cheese tips, and how to cook it
Since rye has a bolder flavor to it, you'll probably want something a little more intense than a mild cheese (so, skip the mozzarella); you'll obviously want cheese that melts well, so avoid the sharp, aged cheeses, as they can become oily. Gruyere is a solid candidate with a rounded nuttiness but a distinct flavor that'll hold up against rye. Gouda is a good option albeit a little mild, so smoked gouda is a good bet. Otherwise, you can't go wrong with classic cheddar. Creamy havarti is a good melter: It's on the mild side, but it often has added extras like pepper or horseradish to add more flavor.
Speaking of add-ins, think about some extras beyond cheese: A sharp cheddar may pair nicely with apples for an intriguing sweet-savory combination. Something acidic like tomato is a good idea to balance out the fattiness of the cheese, or to channel the feeling of a deli-style sandwich. Pickles add a bold flavor that stands up to rye: Anything from classic dill pickles to spicy banana peppers can work. Of course, you can always mix different cheeses: Cheddar and gruyere balance each other out well.
When it comes to cooking, rye doesn't need a different treatment from other breads: Butter the outside and cook it on medium heat or lower so the bread won't burn before the cheese can melt. One final tip: Mayo has a higher smoke point than butter, so if you like your grilled cheese well-done, spread that on the outside instead.