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Wipe outside of chilis with damp paper towel to remove any dust. Place chilis in large heat-safe bowl. Add boiling water, cover and let stand 15 minutes until softened. Transfer chilis and soaking liquid to blender container. Add remaining Sauce ingredients. Cover. Blend until smooth.
Place beef in large Dutch oven or oven-safe saucepan. Add Sauce, stirring to coat beef. Cover and refrigerate at least 2 hours or overnight.
Preheat oven to 350°F. Remove Dutch oven from refrigerator. Add cinnamon stick. Transfer to oven.
Braise, covered, 2 1/2 to 3 hours, or until meat is very tender. Discard cinnamon stick. Skim fat from cooking liquid; transfer to shallow dish or pie plate and set aside. Transfer beef chunks to large bowl and coarsely shred using two forks. Add 1 cup of sauce from Dutch oven to bowl with beef, stirring to coat well. Transfer remaining sauce to bowl for serving. (If you prefer a thinner consistency sauce for dipping, stir in hot water or beef stock, to taste.)
To assemble tacos, heat large non-stick skillet on medium heat. Dip one side of each tortilla into pie plate with reserved fat to coat lightly. Place fat-side down in skillet. Sprinkle tortilla with 1/4 cup cheese and layer with about 1/4 cup of the shredded beef. Cook just until cheese is melted. Sprinkle with cilantro, drizzle lightly with lime juice and fold tortilla in half to enclose filling. Cook just until browned on both sides. Repeat with remaining tortillas and filling. Serve with additional lime wedges and remaining Sauce for dipping.
Test Kitchen Tip: Make quick and easy pickled onions to serve with Birria Tacos. Mix 1 cup fresh lime juice (from about 12 limes), 1 tablespoon kosher salt and 1 tablespoon sugar in tall jar or plastic storage container. Add 1 large red onion, thinly sliced, making sure to completely submerged in brine. Cover. Let stand at room temperature at least 30 minutes or until ready to serve. Refrigerate any leftovers in an airtight container.