Toast the dried ancho, guajillo, and arbol chiles in a large Dutch oven over medium heat until they are fragrant and slightly darkened, about 3 minutes. Transfer them to a bowl and cover with boiling water; let them soak until soft, about 20 minutes.

Pulse the mushrooms in a food processor until they are broken down into pieces no larger than 1/4 inch — the processor does the heavy lifting for the texture here.

Remove the chiles from the soaking liquid and blend with some of the reserved liquid until a smooth paste forms.

Heat the oil in the Dutch oven over medium-high heat until it shimmers. Add the mushrooms and onion and cook until the liquid has evaporated and the mushrooms are well-browned, about 12 minutes.

Stir in the garlic, jalapeño, tomato paste, cumin, and oregano. Cook until the tomato paste is fragrant and has darkened to a rusty brown color, about 2 minutes.

Add the chile paste, tomatoes, beans, soy sauce, bulgur, walnuts, and sugar. Break up the whole tomatoes with a wooden spoon and stir to combine all ingredients.

Bring the chili to a simmer, then reduce the heat to low. Cover and cook until the bulgur is tender and the liquid has thickened into a rich sauce, about 45 minutes.

Season with salt and pepper to taste. Serve hot with lime wedges, fresh cilantro, and sour cream.