Step 1

Heat the oven to 350°F with a rack in the lower-middle position. In a large Dutch oven, stir together the beef, onion chunks, 1 teaspoon salt and 2 cups water. Cover, transfer to the oven and cook until a skewer inserted into the meat meets no resistance, about 3 hours.

Step 2

While the beef cooks, in a 12-inch skillet over medium, toast the garlic and all of the chilies, turning occasionally with tongs, until the garlic is browned on all sides and the chilies are fragrant and just a shade darker, about 5 minutes. Transfer to a medium bowl. When cool enough to handle, break the chilies (or snip them with kitchen shears) into smaller pieces, discarding any remaining seeds, and add them to a blender. Add the boiling water, garlic, bay, cumin, oregano, allspice, thyme and ½ teaspoon salt. Puree, scraping the blender as needed, until smooth, about 1 minute; cover and refrigerate ready to use.

Step 3

When the beef is done, remove the pot from the oven. Using a slotted spoon, transfer the beef to a large bowl. Tilt the pot to pool the cooking liquid to one side, then use a wide spoon to skim off and discard as much fat as possible from the surface. Bring the liquid to simmer over medium-high and cook, stirring occasionally, until reduced to about ½ cup, 5 to 7 minutes; remove from the heat. Meanwhile, shred the beef.

Step 4

To the reduced cooking liquid in the pot, add the chili puree. Bring to a simmer over medium, then stir in the beef. Cook, stirring occasionally, until the mixture begins to sizzle, 5 to 8 minutes. Off heat, stir in the lime juice, then taste and season with salt and pepper.

Step 5

To serve, put the avocados, sliced onion and lime wedges into small bowls. Transfer the birria to a serving bowl, then serve with the tortillas and garnishes for making tacos.