Heat a large non-stick skillet over medium heat. Once the skillet is hot, add the beef and cook, breaking up the meat with a wooden spoon, until browned and cooked through, about 8 minutes.
Meanwhile, in a food processor, combine the bell pepper, onion, jalapeño, and garlic. Pulse until everything is finely minced, 10 to 12 times.
Once the beef is browned, reduce the heat to medium. Add 1 tablespoon of the butter along with the minced pepper-onion mixture. Cook, stirring occasionally, until the liquid is reduced and the vegetables are softened, about 5 minutes. Add the ketchup, water, tomato paste, brown sugar, chili powder, mustard, granulated garlic, Worcestershire, and salt, stirring to combine. Bring to a simmer, then reduce the heat to medium-low. Cook until the sauce is thickened, about 5 minutes more.
Meanwhile, turn the broiler to high. Spread the insides of the hamburger buns with the remaining 3 tablespoons butter. Place the buns, buttered side up on a baking sheet and broil until golden brown, 1 to 2 minutes.
Divide the sloppy joe mixture on the buttered buns. Serve warm.
