Ultimate Grilled Cheese

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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