Preheat the oven to 375 degrees. Place 2 tablespoons of butter in a large skillet and place over medium heat. Once the butter has melted, add the carrot, onion and green bell pepper and cook until slightly softened, about 4 minutes. Add the garlic and cook 1 more minute. Add the ground beef, breaking up into little pieces with a wooden spoon, and cook until no pink pieces remain, 5-7 minutes. If a lot of liquid or fat remain in the skillet, drain most and discard. Add the ketchup, brown sugar, chili powder, dry mustard, beef stock, tomato paste, Worcestershire sauce, salt and pepper. Stir to combine and simmer for 3-5 minutes. Add hot sauce if desired. Transfer the mixture to a 9-by-13-inch casserole dish. Melt the remaining 2 tablespoons of butter. Nestle the slider rolls into the sloppy joe mixture and brush all over with the melted butter. Sprinkle the tops all over with everything bagel seasoning or sesame seeds. Bake for 8-10 minutes until the tops of the buns are toasty. Serve in serving bowls immediately.