Preheat a seasoned grill pan or griddle over medium heat. Assemble this sandwich just as you would a traditional grilled cheese sandwich: spread 1 slice of bread with some Red Onion Mayo, then top with 2 slices Cheddar, the bacon, the apple slices, then 2 more slices of Cheddar, finishing with another mayo-slathered slice of bread. Butter the outsides of the bread and transfer to the hot grill. Grill about 3 minutes per side, since this sandwich is so thick. Remove to a cutting board and cut the sandwich in half on the diagonal to serve.
Preheat oven to 375 degrees F.
On a small rimmed baking sheet, toss the red onion, olive oil, salt and pepper together. Roast in the oven for 25 to 30 minutes until the onion is very soft, stirring occasionally. Remove from oven and let cool. Transfer to a food processor and buzz until finely chopped. Add the mayo and pulse until smooth and combined.