Preheat the oven to 400 degrees. Arrange the bacon on a baking rack set over a sheet pan in a single layer and roast for 20 to 30 minutes, until nicely browned. Drain on a plate lined with paper towels and cut in 1-inch pieces.
Meanwhile, combine the mayonnaise, mustard, Parmesan, salt, and pepper in a small bowl. Lay 12 slices of bread on a board and spread each one lightly with butter. Flip the slices and spread each one generously with the mayonnaise mixture. Don't neglect the corners!
Grate the cheeses in a food processor fitted with the largest grating disk and combine. Distribute the bacon evenly on half the slices of bread. Pile 1/3 cup grated cheese evenly on top of the bacon and top with the remaining bread slices, sauce side down.
Meanwhile, heat an electric panini press. When the press is hot, cook the sandwiches for 3 to 5 minutes in batches until the bread is toasted and the cheese is melted. Allow to cool for 2 minutes. Cut in half and serve warm.
If you don't have a panni then cook on the griddle and weight down with a plate.