Preheat the oven to 400 degrees F. 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.