Prepare the pimento cheese by combining the shredded sharp Cheddar cheese, mayonnaise, Cayenne pepper, and paprika together in a small bowl. Stir to combine. Set to the side while you make the burger.
In a medium bowl combine the ground chuck, salt, and pepper. Divide the meat into 4 portions. Shape each portion into a disk about 1 inch thick, press a small dimple into the middle of the pattie. This dimple will help keep your burgers from shrinking when cooking.
Heat an iron skillet over medium heat. Add 1 teaspoon of butter to the pan and toast the buns on the pan. The average skillet will allow you to heat up 2 buns (2 tops and 2 bottoms), at one time. Repeat this process until all buns are toasted.
Turn the heat up to medium high, place the burger patties onto the skillet. Cook the burgers for 3 to 5 minutes on the first side. The burgers will be ready to flip when you start to see the patty brown on the underside. Flip the patties, and cook for an additional 3 to 5 minutes on the other side.
Building the burger
To build the burger, place the burger patties onto the bottom buns, top the burger patty with one or two tablespoons of the pimento cheese, and then top with the top bun.