How to Make It
Heat olive oil in a large skillet over medium-high. Add onion; cook, stirring often, until softened, about 5 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute. Add ground beef, and cook, stirring to crumble, until lightly browned, about 8 minutes. Push beef mixture to one side of skillet. Add tomato paste, chili powder, salt, and cayenne to empty side of skillet, and cook, stirring constantly, until fragrant and mixture has darkened, 1 to 2 minutes.
Stir in beer, brown sugar, and Worcestershire sauce, incorporating beef. Return to a simmer over medium-high, and cook, stirring occasionally, until liquid has almost completely evaporated, 5 to 7 minutes. Stir in tomato sauce, and bring mixture to a simmer. Reduce heat to low, and simmer, stirring occasionally, until flavors meld and chili has thickened slightly, 15 to 20 minutes. (If chili becomes too thick, add water, 1 tablespoon at a time, to loosen.) Remove from heat, and stir in vinegar. Cover to keep warm.
Preheat a grill to medium-high (400°F to 450°F), or heat a grill pan over medium-high. Grill hot dogs on oiled grates, uncovered, turning occasionally, until heated through and lightly charred, about 6 minutes. Place hot dogs in buns, and top each with about 1/4 cup chili. Reserve remaining chili for another use. Serve with desired toppings.