1. Put meat on a baking sheet, pat dry, then rub all over with 2 tsp. salt.
2. Bring 3 cups water to a boil in a large saucepan; cover and remove from heat. Meanwhile, stem chiles. Heat a cast-iron skillet or heavy frying pan over medium-high heat. Toast chiles in batches, pressing flat against pan's surface with tongs and turning to toast evenly, until lightly browned and fragrant about 2 minutes per batch. Add chiles to hot water as they're toasted and soak, covered, until soft, about 20 minutes.
3. Toast garlic cloves in same frying pan, turning until browned all over. When cool enough to handle, peel.
4. Transfer chiles to a blender (reserve liquid) and whirl with garlic, vinegar, and piloncillo. Grind cumin, oregano, and cloves in an electric spice blender or in a mortar, then add to blender with 1/4 cup chile-soaking liquid. Whirl to make a thick paste, adding a little more soaking water if needed.
5. Smear paste over meat, cover, and chill at least 8 hours and up to 1 day.
6. Preheat oven to 325°. Put a metal cooling rack in a roasting pan (large enough to hold meat in a single layer) and pour in 3 cups water. If water comes above rack, elevate rack with cookie cutters. Arrange meat on rack. Cover pot with a big sheet of heavy foil and crinkle foil around edge to seal tightly. Bake until meat is very tender, 4 to 4 1/2 hours.
7. Transfer meat to a rimmed baking sheet and cover with foil. Measure broth, skim off fat, and add enough water to make 3 cups.
8. Whirl tomatoes in a blender until smooth, then whirl in broth. Strain into a saucepan and heat over medium heat until steaming.
9. Divide hunks of meat among 6 soup bowls and ladle about 1/2 cup broth over each. Season with more salt to taste.
*Find cabrito, guajillo and ancho chiles, and piloncillo (Mexican brown sugar) at Latino markets. You can often order cabrito through butcher shops too.
Note: Nutritional analysis is per serving (calculated with lamb; goat data unavailable).