12 slices thick-cut bacon, such as Nodine's applewood smoked
1 cup good mayonnaise
1/4 cup Dijon mustard
1/4 cup freshly grated Parmesan cheese
1 1/2 teaspoons kosher salt
1/2 teaspoon freshly ground black pepper
1 white pullman loaf or sourdough bread, sliced 1/2 inch thick (12 slices)
6 tablespoons salted butter, at room temperature
6 ounces aged Gruyere or Comte cheese
6 ounces extra-sharp Cheddar, such as Cabot or Shelburne Farms