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.