In a heatproof bowl, soak the arbol, guajillo, and ancho chiles in the boiling water until softened, about 20 minutes. Drain the chiles, reserving 1/3 cup of the soaking liquid. Coarsely chop the chiles.

In a small skillet, toast the cumin seeds over moderate heat until fragrant, about 20 seconds. Transfer the seeds to a blender. Add the chiles and their reserved soaking liquid along with the oregano, garlic, paprika, and 1 tablespoon of salt. Puree until smooth. Scrape the chile puree into a large nonreactive bowl or baking dish. Add the goat and toss to coat thoroughly. Cover and refrigerate overnight.

In a large saucepan, cover the beans with 2 inches of water and bring to a boil. Simmer over low heat, stirring occasionally, until tender, about 1 hour; add more water as needed to keep the beans covered by 2 inches. When the beans are just tender but still al dente, season them with salt and let stand in their cooking liquid for 5 minutes.

Preheat the oven to 375 degrees. In a large, enameled cast-iron casserole, cook the bacon over moderate heat until the fat has rendered, about 3 minutes. Using a slotted spoon, transfer the bacon to a large plate. Add the olive oil to the casserole. Working in batches, cook the chile-goat mixture over moderately high heat, turning a few times, until richly browned all over, about 4 minutes. Transfer the browned goat to the plate with the bacon.

Add the onion to the casserole and cook over moderately low heat, stirring occasionally, until softened, about 10 minutes. Add the goat and bacon and any accumulated juices and stir well. Add the beer and boil over high heat until reduced by half, about 8 minutes. Add the chicken stock and bring to a simmer.

Cover the casserole, transfer it to the oven and bake for about 30 minutes, until the goat is tender when pierced with a fork. Add the beans and bake, uncovered, for about 10 minutes, until they are warmed through. Remove the casserole from the oven and let rest for 10 minutes. Season with salt and pepper. Transfer the chili to bowls and serve with the sour cream, cilantro sprigs, and lime wedges.

The chili can be refrigerated for up to 3 days. Reheat gently.