Chile-Braised Beef, Two Ways

Place beef in a large pot and season with 1 tablespoon salt. Add 1 onion half, 5 garlic cloves, bay leaves, and enough water to cover meat by 1 inch (8 to 10 cups). Bring to a boil, skimming impurities from surface. Cover, reduce heat to medium-low, and simmer until tender enough to easily pierce with the tip of a knife, about 2 hours.

Meanwhile, preheat broiler, with rack 6 to 8 inches below heating element. Arrange chiles on a rimmed baking sheet. Broil with oven door cracked open, turning once, until blistered in places on both sides, fragrant, and beginning to smoke, 30 to 45 seconds. Transfer to a saucepan. Arrange tomatoes and remaining onion half and garlic cloves on baking sheet. Broil, turning a few times, until blistered in places and beginning to soften, 8 to 10 minutes.

Transfer to saucepan with chiles; add enough water to just cover vegetables (4 to 5 cups). Bring to a boil, then reduce heat to medium-low and simmer, uncovered, 10 minutes. Strain, reserving 1 cup cooking liquid, and transfer to a blender with reserved liquid, vinegar, cinnamon, cloves, cumin, oregano, and 2 teaspoons salt. Purée until smooth.

Transfer beef to a plate and strain broth through a fine-mesh sieve, discarding solids. Return beef to pot with 5 cups broth and puréed-chiles mixture (any remaining broth can be cooled and stored in an airtight container in the refrigerator up to 1 week, or frozen up to 6 months; if you'll be serving the dish as a soup, reserve and keep warm).

Bring beef to a boil. Partially cover, reduce heat to medium-low, and simmer until beef is fork-tender, 1 1/2 to 2 hours more. Shred meat into bite-size pieces.

To serve as a soup, add reserved warm broth to pot, a little at a time, to reach desired consistency (it should be soupy but still have some body to it). Scatter with chopped onion and cilantro and serve, topped with radishes and avocado, with lime wedges alongside.

To make tacos, dip tortillas into chile broth, flipping to fully coat, and transfer to a plate, stacking them one on top of another. Fill each with about 1/3 cup shredded beef and a little more chile broth. Fold in half. Heat a generous slick of oil over medium-high in a large, heavy skillet, such as cast iron. When it shimmers, add tacos in a single layer and cook, flipping once (if browning too quickly, reduce heat to medium) until crisp and golden brown in places, about 2 minutes per side. Serve with chopped onion, cilantro, hot sauce, and lime wedges.