Add the ground beef and water to a saucepan or skillet. Before turning the mixture into a browned crumble, use a potato masher or wooden spoon to break the raw beef apart as thoroughly as possible.

Add the finely minced onion, garlic, tomato sauce, yellow mustard, chili powder, paprika, Worcestershire sauce, cumin, kosher salt, black pepper, and optional cayenne. Stir until the mixture is evenly combined.

Bring the sauce to a gentle simmer over medium heat, then cook uncovered for 30 to 45 minutes. Stir occasionally and continue mashing any remaining pieces of beef until the sauce is very fine, thick, and still spoonable.

If the sauce becomes too dry before the texture is ready, add a small amount of water and continue simmering. Taste near the end and add more salt if needed.

Steam or simmer the beef hot dogs, or grill them if you prefer a lightly charred finish. For the most traditional soft Coney-style result, steaming or simmering is the usual choice.

Steam or gently warm the buns so they remain soft and flexible. Avoid heavily toasting them, since the tender bun is meant to cradle the sauce.

Place one hot dog in each bun and add a strip of yellow mustard. Spoon the warm Coney sauce over each hot dog, then finish with finely minced raw yellow onion.