Chop one-half of the onion, and slice the other half.
Heat 1 T of the olive oil in a big skillet, over a medium heat, until hot. Reduce the heat to med-low and then add the chopped onion, cook them for about 4-5 minutes, or until soft. Press 1 garlic clove into the skillet and cook for only about a minute. Take out the onion and garlic and reserve.
Heat the remaining olive oil in the same skillet over a medium heat. Add the sliced onion and cook for about 4-5 minutes, stirring. Take those onions out and set aside to cool. Wipe the skillet out.
Mix the mayo and paprika and press the other garlic clove into it and mix.
Add the pork and ground round to the cooled chopped onion mixture and stir. Pat out the meat mixture into 6 burgers. Then season them on both sides with salt and black pepper, to taste.
Heat the remaining olive oil in the skillet, over medium heat. Put the burgers into the hot skillet and cook for about 5-6 minutes per side, or until they are just about cooked through.
While the burgers are cooking, combine the piquillo peppers, olives and the sliced onion and then season with salt and pepper to taste. Divide the mixture on top of each of the 6 almost cooked burgers, while in the skillet. Then top each with a slice of cheese. Put the heat on low, cover the skillet, and cook for about 1 minute. You just want the cheese to melt.
Spread the top and bottom of each hamburger bun with the mayo mixture and then put a burger on the bottom half of each bun and top.