In a large skillet, melt the butter over medium-high heat. Add the ground beef and cook, breaking up the meat with a wooden spoon, until browned, 5 to 7 minutes. Drain off and discard most of the fat.
Add the garlic, bell pepper, onion, salt, and black pepper and cook, stirring occasionally, until the vegetables begin to soften, 5 to 7 minutes. Pour the mixture into a 6-quart slow cooker.
Add the ketchup, brown sugar, Worcestershire sauce, chili powder, dry mustard, red pepper flakes, and ½ cup water. Stir to combine.
Cook until the flavors concentrate and the mixture reduces slightly and deepens in color around the edges, on high for 2 hours or low for 4 hours. Taste and adjust the seasonings as needed.
Heat a griddle or skillet over medium heat. Spread the cut sides of the rolls with butter and, working in batches as necessary, toast buttered-side down until golden brown, 1 to 2 minutes. Spoon the meat mixture (about ½ cup) over each roll, topping with slices of cheese, if you like. Serve with hot sauce, chips, and a pickle.