For the pimento cheese: Add the Cheddars, cream cheese, pimentos and mayo to a bowl and mix together. Add the chives, garlic powder, cayenne, smoked paprika and salt and pepper and mix until thoroughly combined.
Place the cheese mixture on a sheet of plastic wrap and form into a log. Freeze for a few minutes or refrigerate up to overnight.
For the sauce: Mix together the mayonnaise, hot sauce, Worcestershire, paprika and 1 tablespoon reserved pimento juice in a medium bowl. Season with salt and pepper.
For the burgers: Preheat a grill to medium heat.
Mix the meats, breadcrumbs, Worcestershire, egg, salt and pepper together in a bowl with your hands until completely combined. Divide the meat into 8 portions and shape each into large thin patties. Slice four 1/2-inch slices off the pimento cheese log (reserve the rest for another use…snacking!). Place a pimento cheese slice on four of the patties and then top them with the remaining four patties, sealing the sides of the patties to enclose the cheese. (See Cook's Note.)
Lightly oil the grill grates. Add the patties to the hot grill and cook until they reach an internal temperature of 160 degrees F, about 4 minutes per side.
Spread the sauce on the buttered buns. Top each with lettuce, tomato, pickles, onions and a patty and close the burgers. Eat with lots of napkins.
Stuffed patties can be made ahead and refrigerated until ready to use.