In a saute pan over high heat, add the butter, onions, white pepper and 1 teaspoon salt, and cook over high heat until the butter has melted. Reduce the heat to medium and finish cooking the onions until slightly browned on the edges, 7 to 8 minutes.
Mix the beef, pork, remaining 1 teaspoon salt, and black pepper in a bowl. Do not over mix or the burger will be too tightly bound. Portion the meat into 12 (2-ounce) balls, then flatten the balls into patties about 2 1/2 inches round.
Preheat a grill and grill the burgers for 4 to 5 minutes on each side, or until the internal temperature reaches 160 degrees F.
Combine the mayonnaise-mustard mixture with the lemon juice and shredded romaine. Top the bottom bun with the dressed lettuce, a burger, some caramelized onions, and cheese of choice. Add the top bun and serve.