This has become a family favorite since Dad started making it.
- Bring 4 quarts water, 3 tablespoons salt, and beans to boil in large Dutch oven over high heat. Remove pot from heat, cover, and let stand for 1 hour. Drain beans and rinse well. Wipe out pot.
- Adjust oven rack to middle position and heat oven to 300 degrees. Arrange anchos and New Mexican chiles on rimmed baking sheet and toast until fragrant and puffed, about 8 minutes. Transfer to plate and let cool, about 5 minutes. Stem and seed anchos and New Mexican chiles. Working in batches, grind toasted chiles, mushrooms, and oregano in spice grinder or with mortar and pestle until finely ground.
- Process walnuts in food processor until finely ground, about 30 seconds. Transfer to bowl. Process drained tomatoes, tomato paste, jalapeño(s), garlic, and soy sauce in food processor until tomatoes are finely chopped, about 45 seconds, scraping down bowl as needed.
- Heat oil in now-empty Dutch oven over medium-high heat until shimmering. Add onions and 1 1/4 teaspoons salt; cook, stirring occasionally, until onions begin to brown, 8 to 10 minutes. Lower heat to medium and add ground chile mixture and cumin; cook, stirring constantly, until fragrant, about 1 minute. Add rinsed beans and water and bring to boil. Cover pot, transfer to oven, and cook for 45 minutes.
- Remove pot from oven. Stir in bulgur, ground walnuts, tomato mixture, and reserved tomato juice. Cover pot and return to oven. Cook until beans are fully tender, about 2 hours.
- Remove pot from oven, stir chili well, and let stand, uncovered, for 20 minutes. Stir in cilantro and serve. (Chili can be made up to 3 days in advance.)