Combine meat with salt, pepper, paprika and parsley. Form 4 large patties, making them thinner at middle for even cooking and to counteract the bulging of the meat as it cooks. Drizzle burgers with EVOO to coat.
Heat a large, cast-iron skillet over medium-high heat. Cook burgers 8 minutes turning once for medium-rare and up to 12 minutes for medium-well.
In a medium saucepot over medium to medium-low heat, melt butter, then add onions and garlic and cook 5 minutes. Add flour and stir to combine. Add milk and heat through for a minute, then melt the cheese into milk the sauce will be very thick, barely spoonable. Add pimientos, Worcestershire, hot sauce and season with salt and pepper to taste.
Serve burgers on buns with pimiento cheese sauce on top and a pickle and chips alongside.