1Heat the vegetable oil in a large skillet over medium-high heat. Add the ground beef and break it into very small crumbles with a spoon or potato masher. Cook until browned, about 6 minutes.
2Add the onion and cook until soft, about 4 minutes. Stir in the garlic, chili powder, cumin, and smoked paprika and cook 30 seconds until fragrant.
3Stir in the tomato paste, tomato sauce, water, Worcestershire, vinegar, and brown sugar. Simmer over medium-low heat, stirring now and then, until thick and spoonable, about 12 minutes. Season with salt and move the chili to a bowl or keep warm on the lowest heat.
4Wipe out the skillet and set it over medium-high heat. Add the hot dogs and cook, turning, until blistered and heated through, about 5 minutes.
5Toast the buns cut-side down in the same skillet for 30 to 60 seconds, or until golden.
6Tuck a hot dog into each bun, spoon over a generous scoop of chili, and top with shredded cheddar and a squiggle of mustard. Serve right away with napkins.