For Paul's burger sauce: In a medium bowl, whisk together all of the sauce ingredients.
For the burgers: Heat a large cast-iron skillet over medium-high heat until almost smoking. Add about 1 tablespoon of oil and 1 tablespoon of the butter to the skillet. When the butter has melted, working in batches, open the buns and toast them in the skillet until they are golden brown.
Wipe out the skillet and reheat it over medium-high heat. Season the burger patties with salt and pepper. Working in batches and using more of the oil and butter, place some of the burger patties in the skillet (do not crowd the pan). Cook until the burgers are well seared (about 3 minutes) and flip. Place a slice of cheese on each burger and continue to cook the burgers for 2 minutes longer. Cover the skillet to melt the cheese and steam the burgers. Continue until all the burgers are cooked, adding more oil and butter to the skillet between batches as necessary.
Spread each side of the buns with about 2 teaspoons of Paul's burger sauce. Season the tomato slices with salt and pepper and place the tomatoes on the bottoms of the buns. Place the cooked burger patties on top of the tomatoes and then top the burgers with shredded lettuce. Secure the tops of the buns on the burgers with frilled toothpicks or small bamboo skewers.