Schmear the inside of the bagel halves with mayo or butter. Flip one side over and layer cheese over the bagel's top, avoiding the hole. Add fillings, then add more cheese. Place the second bagel half, cut side up, on top of the cheese. Press down with a spatula.
Heat a large nonstick frying pan or griddle over medium-low heat until hot. Transfer the bagel-cheese-bagel situation to the hot pan and cook until the cheese starts to melt and the bottom is brown, about 4 minutes. Press the sandwich down a bit with a spatula.
Then gingerly flip the sandwich with the spatula and let the other side brown.
Once the cheese is totally melted, transfer the sandwich to the oven to get a bit crispier and meltier. Eat!