Heat a gas or charcoal grill to high heat. Brush the grill grates with oil.
For the fry sauce: Combine the mayonnaise, roasted red pepper puree and adobo sauce in a small bowl and stir together until smooth. Season with salt and pepper to taste. Use right away or cover and refrigerate for at least 1 hour to allow flavors to meld.
For the burgers: Divide the meat into 4 equal portions (about 6 ounces each). Shape each portion with your hands into a round patty, about 3/4 inches thick. Use your thumb to make a divot in the center of each burger. Sprinkle both sides of each burger with salt and pepper.
Grill, covered, until golden brown and slightly charred on the first side, about 3 minutes. Flip the burgers and cook until golden brown and slightly charred on the second side, about 4 minutes for medium.
While the burgers cook, toast the buns on the grill until lightly charred, 10 to 15 seconds. Set aside.
Add 2 slices of cheese to the tops of each burger. Cover the burgers with the grill lid, large domed lid or large metal bowl and count to 10 until the cheese is melted.
Spread some yellow mustard on the bottom buns and spread some of the fry sauce on the top buns. Place the burgers on the bottom buns. Top with the shredded lettuce and the top buns and serve immediately.