In the bowl of a stand mixer, combine the mascarpone, piquillo pepper puree, mayonnaise, 1 tablespoon salt, smoked paprika, and white pepper. Mix on low speed until all of the ingredients are incorporated evenly. Remove the bowl from the mixer and add the sharp Cheddar cheese, diced piquillo peppers, aged sharp Cheddar cheese, Parmesan cheese, and parsley to the bowl. Mix by hand until evenly incorporated. Adjust seasoning, if needed.
Place the cheese into a container and refrigerate for 1 day.
Lay the ground beef on top of a large sheet of plastic wrap. Place another piece of plastic wrap on top of the beef. Press the beef out to form a large square about 13 inches on all sides and about 3/4 to 1-inch thick.
Lay a piece of the pullman bread on top of the plastic and cut burgers slightly larger than the bread. Cut a total of 4 burgers all the same size. Remove the plastic on both sides of the meat and season with salt and pepper on both sides. The burgers can be made and seasoned up to a day in advance.
Place a large cast iron skillet over high heat and allow it to heat about 4 to 5 minutes. If you don't have one large enough for all the burgers use 2 skillets. When the skillet is hot, reduce the heat to medium-high and add the burgers to the pan. Cook on the first side for 3 to 4 minutes, or until it has a nice crust on the bottom.
Add the butter to the pan. Flip the burgers over and cook for another 4 to 5 minutes. Remove the burgers from the pan and keep warm. Allow the meat to rest for 3 to 4 minutes. Add the brioche and brown in the beef fat. Add pimento cheese to the bread and allow it to melt while it is browning.
Add the rested burgers to the browned brioche top with the pimento cheese. Cut the burgers in half and serve.
Restaurant Recipe: This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.