For the chili: Heat the oil in a large pot over medium-high heat. Add the onion and cook, stirring frequently, until soft and translucent, about 8 minutes. Add the beef and garlic and cooking, stirring and breaking up into smaller pieces with a wooden spoon, until the meat is brown and cooked through, about 5 minutes. Add the chili powder, cayenne, cumin, 1/2 teaspoon salt and a few grinds of pepper, cook, stirring, until well combined, about 1 minute. Add the tomatoes, tomato paste, and 1/2 cup water to the pot. Stir well and bring to a boil. Lower the heat to medium-low and simmer, uncovered and stirring occasionally, until slightly thickened, about 10 minutes. Season with salt and pepper and keep warm.
For the chili dogs: Preheat the grill to medium. Grill the hot dogs according to the package directions. Toast the cut side of the buns on the grill until golden. Build the hot dogs and top each with about 1/4 cup of chili spooned down the length of the hot dog. Sprinkle each with some cheese and serve with ketchup and mustard.
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