Toss the Taleggio in the freezer for 20 minutes to get it really nice and firm (don't forget about it!). Remove the orange rind and cut the cheese into slices about 1/4 inch thick.
While the cheese is in the freezer, cook the bacon in a large saute pan over medium heat until brown and crispy on both sides, then transfer it to a plate lined with paper towels and reserve.
Spread each piece of bread with a thin, even layer of mayonnaise or butter. Spread the other side of each piece of bread with Dijon. Lay the bread down on a baking sheet with the Dijon side facing up. It will be a bit messy. This is what happens with grilled cheese! The mess is worth it.
Place an even layer of Taleggio on each piece of bread. Lay 3 bacon slices each on 4 pieces of the Taleggio-lined bread. Top the bacon with an even layer of apple slices (these are the bottoms, the ones with just cheese are the tops—together they make a pair!).
Preheat a griddle or nonstick saute pan to medium. You will most likely have to work in batches, so also preheat the oven to 200 degrees F to keep the finished sandwiches warm. Place pairs of tops and bottoms (not yet sandwiched together) on the griddle or in the nonstick pan and cook until the underside gets brown and crusty and the cheese begins to melt, 5 to 7 minutes. Flip the tops onto the bottoms so the cheese is on the inside. Give each sandwich a press, flip it over, and cook for 1 to 2 minutes more. Keep the finished sandwiches in the oven and repeat the process with the remaining tops and bottoms.