Mix the sherry vinegar and sugar in a small bowl until the sugar is dissolved.
Heat the oil in a 1-quart saucepan over medium heat. Add the shallot, serrano, and a pinch of salt and cook, stirring occasionally, until the shallot is tender and translucent, 3 to 4 minutes. Add the roasted red peppers and cook for 1 minute. Add the vinegar mixture and simmer until the liquid is mostly evaporated, 7 to 9 minutes. Stir in the thyme, season to taste with salt and pepper, and let cool to room temperature.
Position a rack 6 inches from the broiler element and heat the broiler on high.
Put the ground beef in a mixing bowl and add 1/2 cup of the relish, 1 tsp. salt, and 1/2 tsp. pepper. Mix gently and briefly to avoid overworking the meat. Shape the beef into nine 1/2-inch-thick patties.
Lightly oil a broiler pan. Arrange the burgers on the pan and broil until lightly browned, 2 to 3 minutes. Flip the burgers and broil until lightly browned and cooked to medium doneness (140°F), about 2 minutes.
Serve the burgers in the rolls, topped with the remaining relish.