To make the mayonnaise: combine the mayonnaise, mustard, garlic, anchovies, lemon juice, Tabasco, Worcestershire, and pepper in a food processor; process until smooth.
Scrape into a bowl and stir in the cheese; cover and refrigerate at least 30 minutes or up to 8 hours before serving so flavors can meld.
To make the burgers: divide the meat into 4 equal portions (6 oz each); form each portion loosely into a ¾ inch thick burger and make a deep depression in the center with your thumb.
Season both sides of each burger with salt and pepper; cook the burgers, using the oil-- heat oil in the pan or griddle over high heat until the oil begins to shimmer; cook the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef and 5 minutes for turkey.
Flip burgers over; cook beef burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare or until cooked to desired degree of doneness; cook turkey burgers until cooked throughout, about 5 minutes on the second side.
Place the burgers on the bun bottoms and spread a few tablespoons of the Caesar mayonnaise on each burger.
Top with the romaine lettuce leaves and sprinkle with the Parmesan and black pepper.