Bring a teakettle of water to a boil. In a cast iron pan over medium heat, toast the chiles on each side until fragrant, about 5 minutes. Remove from the pan, place in a heatproof bowl, and cover with boiling water. Let sit for 20-30 minutes. Drain the chiles, de-stem and seed them, then add to a blender.
Meanwhile, grind the allspice, cumin, and oregano. Add to the blender along with the thyme, onion, garlic, vinegar, and 1/2 cup water. Season the goat all over with plenty of salt and pepper.
Put into a roasting pan, and rub the chile paste all over the goat. Cover the pan tightly with aluminum foil and refrigerate overnight.
(Note: we only ended up using about half of the paste, and reserved the other half to marinade something else, like a chicken. It would also probably be great on lamb.)
Preheat the oven to 300F. Cook the goat for about 4-1/2 to 5 hours, turning once about halfway through, until very tender and falling off the bone. Pour off the juices into a bowl or glass measuring cup and skim off the fat. Shred the meat. Pour the juices back over the meat and toss to combine.
Serve with corn tortillas, pickled onions, and crema.
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