Combine the sun-dried tomatoes, basil, parmesan, peppercorns, bay leaves, and milk in a heavy-bottomed saucepan. Place the pot over medium-high heat and bring to a boil. Remove the pot from the heat and place a lid on the pot. Set aside to infuse for at least 15 minutes and up to 2 hours.
After infusing, pour the mixture through a fine mesh strainer and reserve the milk, discarding the solids.
Warm the Basil Parmesan milk and set aside.
In a separate pot, melt the butter over medium heat. When the butter foams, add the flour and cook, stirring often until it forms a thick paste the color of toasted almonds. Gradually whisk the warmed milk into the pot and bring to a simmer, stirring constantly. Reduce the heat and cook until thickened, about 2 to 3 minutes. Season the finished béchamel to taste with salt and pepper.
Transfer 1/2 cup of the thickened milk to a bowl and stir to cool slightly. Add the grated cheeses to the bowl and mix well. Reserve the remaining béchamel.
Lay out the slices of bread and spread the cheese mixture on one side of each piece of bread.
On top of the cheese mixture on 4 of the slices of bread, stack the remaining in the following order: tomato, prosciutto, salami, capocolla, pepperoncino. Then, top each with a piece of bread spread with cheese.
Generously butter the outside of each sandwich and brown in a cast iron or non-stick sauté pan. Cook the sandwiches over medium heat until well browned on both sides, then transfer to a rack set over a baking pan.
Spread enough of the béchamel to completely cover the top face of each of the sandwiches—you may not use all of the béchamel. Sprinkle each sandwich with grated parmesan cheese and place the pan underneath a preheated broiler until golden. Cool slightly before serving.