For the ketchup: Heat a grill or grill pan to medium-high heat. Grill the tomatoes skin-side down until dark grill lines appear, about 5 minutes, and then flip and cook until nicely charred and sagging, about 10 more minutes. Remove, chop into pieces, and transfer to a medium saute pan over medium heat. Stir in the vinegar, sugar, salt, aji amarillo paste, allspice, bay leaf and cinnamon stick and cook, stirring occasionally, until the mixture looks very dry and the tomatoes no longer hold their shape, 25 minutes. Discard the cinnamon stick and bay leaf, and then blend the mixture to a smooth, thick sauce about the consistency of store-bought ketchup. If the ketchup is too thick, add 1 tablespoon water at a time and blend until desired consistency is reached.
For the burgers: Sprinkle the onions and peppers with 1/2 teaspoon salt and toss with 1 tablespoon of the oil. Grill the onions and peppers until well-charred and softened, flipping once, about 6 minutes. Remove and set aside. Wipe off the grill to cook the burgers.
Divide the ground beef into four 6-ounce patties, 4 inches wide and 1-inch thick. Brush the burgers with the remaining oil and sprinkle both sides with salt and black pepper. Grill the burgers until visibly cooked more than halfway up the bottom, about 4 minutes, then flip and grill 4 minutes longer for medium rare. Transfer to a cutting board or plate to let rest for 5 minutes.
Place the burgers on the buns and top each with 2 tablespoons of the ketchup, 3 or 4 pepper strips and 4 or 5 onion rings. Serve.
NotesCook's Note: Save any extra ketchup in the refrigerator and use for other recipes, such as grilled pork chops.