Make the marinade. In a large (10-inch) skillet over medium heat, toast the chiles a few at a time, using a metal spatula to press them flat against the hot surface until they change color slightly and release their aroma, then flipping and toasting the other side. Collect the toasted chiles in a bowl, cover with hot tap water, weight with a plate to keep them submerged and rehydrate for 20 to 30 minutes. While the chiles are soaking, toast the garlic in the skillet over medium heat, turning occasionally until soft and blackened in spots, 10 to 15 minutes. Cool and peel.
Drain the chiles, reserving about a cup of the soaking liquid. In a blender or food processor, combine the chiles, reserved liquid, peeled garlic, cumin, pepper, oregano and vinegar. Process until smooth. Taste and season highly with salt, usually a generous teaspoon, and just enough sugar to bring out the fruitiness of the chiles.
Cook the birria. To cook the birria in the oven, turn on the oven to 325 degrees and position the rack one-third up from the bottom. You need a 6-quart pot or Dutch oven with a tight-fitting lid. Lay the meat in the pot (if the goat is too big, use a cleaver to cut it in half), pour the marinade evenly over it, then pour 4 cups water around the meat. Set the lid in place and cook for 3 hours. To cook the birria in a 6-quart slow-cooker, follow the directions above, but reduce the amount of water to 1 cup. Cook on high for 6 hours.
Finish the birria. Use tongs to carefully pull out the meat, laying the pieces on a plate or rimmed baking sheet. Carefully spoon off all the fat into a bowl. Measure the braising liquid. If it’s more than 2 cups, set the pot or Dutch oven over medium-high heat and briskly simmer the delicious broth until reduced to that quantity. Taste and season with salt, usually about 1 ½ teaspoons. Pull the meat into large shreds.
Make tacos. To have the full birria taco experience, you need to work on a large griddle—the kind that covers 2 burners. Set it over medium to medium-high heat. In a very large (12-inch) skillet over medium-high heat, pour in just enough of the reserved fat to coat the bottom. Add the meat and let cook undisturbed until it just begins to brown underneath. Stir to bring uncrusted meat in contact with the skillet, then reduce the heat to its lowest setting. On the griddle lay out a single layer of 4 to 6 tortillas, first dipping each one about 1/3 into the reserved fat, then rubbing the torilla around on the griddle to ensure that its entire bottom is lightly coated with fat. Top them with some of the meat, splash with a little of the reduced broth, sprinkle with onions and cilantro, and sprinkle with a little salt. When the tortilla just starts to brown a little and crisp (don’t wait too long), fold them over and transfer to a warm plate.
Serve the tacos. Serve your just-made tacos with lime wedges and salsa, plus, if you wish, a small cup of the reduced broth for your guests to dip into.