To start the Coney sauce, in a large pot or skillet, heat the vegetable shortening over medium heat.
Add the ground beef. As it cooks, use a potato masher to crumble it into fine pieces.
Once the beef is crumbled and browned, stir in the cracker crumbs.
Add the onion and garlic and cook until fragrant, about 2 minutes.
Mix in the tomato paste and cook until the contents begin to stick to the bottom of the pot, about 2 minutes.
Put in the Worcestershire sauce, mustard, chili powder, paprika, salt, cumin, cayenne, allspice, and beef broth. Stir to combine and bring to a simmer.
Cover the cookware with a lid, reduce the heat, and simmer for 1 hour, stirring occasionally and adding broth as needed to prevent burning, before removing from the heat.
When the Coney sauce is almost done simmering, begin making the hot dogs by heating a grill pan or cast iron skillet over medium heat and adding the vegetable shortening.
Place the hot dogs spaced out on the pan or skillet.
Rotate the hot dogs often until browned, about 10 minutes.
Warm the hot dog buns in the microwave for 10 seconds. Place the hot dogs in the buns and top with Coney sauce.
Garnish with diced onions and a drizzle of yellow mustard to serve.