Heat the olive oil in a medium-sized pot or pan over medium heat.
Add the ground beef to the pot and cook until fully browned, breaking the meat into small crumbles as it cooks.
Drain any excess liquid or grease from the cooked ground beef.
Add the finely chopped onion to the browned ground beef.
Stir the onion into the meat and cook for several minutes, or until the onion begins to soften.
Add the chili powder, ground allspice, ground mustard, garlic powder, ground cumin, ground cinnamon, freshly ground black pepper, and cayenne pepper.
Stir until the spices are evenly distributed throughout the ground beef and onion mixture.
Add the tomato paste and salt.
Stir thoroughly until the tomato paste is fully incorporated into the meat mixture.
Pour in the water and stir until the mixture is evenly combined.
Reduce the heat to low and bring the sauce to a gentle simmer.
Simmer the sauce for 1 to 2 hours, stirring occasionally to prevent sticking.
For a looser Coney-style sauce, simmer for approximately 1 hour.
For a thicker sauce, continue simmering closer to 2 hours, adding a small amount of water if needed to maintain a spoonable texture.
Taste the sauce and adjust the salt, black pepper, or cayenne pepper as desired.
Cook the hot dogs according to package directions or preferred method.
Warm or lightly toast the hot dog buns, if desired.
Place each cooked hot dog into a bun.
Spoon the warm Signature Coney Hot Dog Topping over each hot dog.
Garnish with finely chopped onions and prepared yellow mustard.
Add shredded cheddar cheese, if desired.
Serve immediately while warm.