Blend together the sirloin and short ribs and sprinkle with salt and pepper. Loosely form the meat into 4 equal 4-ounce patties about 1/2-inch thick. I like to shape my patties into either squares or circles depending on the shape of my bread. That way every bite will have burger in it! Create a small well in the center of each beef 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 beef patties generously 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.
Homemade Pimento Cheese:
Blend the Cheddar, mayonnaise, onions, pimientos, garlic and hot sauce in a food processor until well combined. Transfer the pimento cheese to a glass jar or bowl and refrigerate until firm, at least 1 hour.