Create a small well in the center of each Rocket Patty, using your thumb (this will help your burger cook evenly).
Heat 2 tablespoons of the oil in a small saute pan over medium heat. Cook the onions, sprinkling with salt and stirring occasionally, until caramelized, brown and soft, 25 to30 minutes.
Season patties with salt and pepper. Heat a large cast-iron skillet or griddle pan over high heat and add 1 to 2 tablespoons of the oil. Cook the burgers well-side up (the burger well collects the juices while cooking, letting your burger marinate) until browned on each side, 3 to 4 minutes per side. Transfer to a plate.
Spread the Homemade Pimiento Cheese on the non-buttered side of each slice of bread. Top half of the bread slices with a pile of onions, then add the burger patties and top with the remaining bread slices.
Place the sandwiches back on the griddle and toast until the bread is golden brown and the cheese is beginning to melt, 2 minutes per side.