Place ground beef and onion in a large skillet over medium heat; cook and stir until beef is browned and crumbled, about 10 minutes.
Stir in green pepper and garlic; cook and stir until softened, 2 to 3 minutes. Add 1 cup water and stir, scraping the pan to dissolve any brown flavor bits from the bottom of the skillet.
Stir in ketchup, brown sugar, Dijon mustard, salt, black pepper, and Worcestershire sauce. Add remaining 1 cup water and bring mixture to a boil. Reduce heat to low and simmer, stirring occasionally, until liquid has evaporated and mixture is thick, about 40 minutes.
Season with salt, black pepper, Worcestershire sauce, and cayenne pepper.