Roast the pepper under the broiler or over a gas flame, turning it occasionally, until charred all over. Place the pepper in a bowl, cover tightly with plastic wrap and let steam until cool, about 15 minutes. Peel, stem and seed the pepper and place it in a food processor. Add the ketchup and vinegar and puree until the sauce is smooth.
In another bowl, combine the lamb, pepperoni, scallions, herbs, garlic and olive oil. Form into 4 patties (thinner in the centers for even cooking). Heat a cast-iron skillet over medium-high. Cook the patties until browned, turning once, about 8 minutes. Top with the cheese; cover and cook until the cheese melts, about 1 minute.
Mound a handful of greens on each bun bottom, then top with the patties. Slather the bun tops with the roasted-pepper ketchup and place on top of the patties.