To make the mayo, preheat the oven to 425°F and line a shallow baking pan with parchment paper. Cut the red onion into 1/2-inch thick slices and arrange on the parchment. Brush with vegetable oil and season with salt and pepper. Roast until the onion is tender and caramelized, 10 to 12 minutes. Set aside and allow to cool.
While the onions roast, fry the bacon in a heavy, ovenproof, skillet until cooked through, but not completely crisp. Drain on paper towels and allow to cool. Wipe the excess fat from the pan.
Place the ground beef, bacon, smoked paprika, sherry vinegar, 1/4 teaspoon salt and a few grinds of black pepper in a large bowl. Using your hands, combine thoroughly and form into 4 equally sized patties about 1/2-inch thick.
Heat the skillet to medium-high (add a touch of oil if needed) and sear the burgers, 1-1/2 minutes per side. Top each burger with a slice of cheese and place the skillet in the oven until the cheese melts, and the burgers reach the desired doneness (about 1-1/2 minutes additional for medium). Transfer to a plate and allow to rest for 2 minutes.
To finish, chop the roasted onion into small pieces and place in a small bowl. Stir in the mayo, smoked paprika and a pinch of salt and pepper. Divide the mixture between the 4 rolls, top with a burger and serve immediately.