Steps:
- Bring chicken, water, 2 onion quarters, and 1 teaspoon salt to a boil, covered, in a 4- to 5-quart pot over moderately high heat. Boil 10 minutes, then remove from heat and let stand, covered, until chicken is just cooked through, about 10 minutes. Transfer chicken to a plate, reserving broth with onion. When cool enough to handle, coarsely shred chicken, discarding skin and bones.
- While chicken is cooking, heat a dry well-seasoned small cast-iron skillet over moderate heat until hot, then brown garlic and remaining 2 onion quarters on all sides, turning with tongs, about 5 minutes. Peel garlic and transfer with onion to a blender. Add tomatoes with juice, chiles, and oregano, then purée until smooth.
- Cook chorizo in oil in a 12-inch heavy skillet over moderately high heat, stirring, until fat is rendered, about 2 minutes. Carefully add purée (it will splatter and steam) and cook, stirring frequently, until thick, about 10 minutes.
- Peel potatoes and cut into 3/4-inch pieces, then add to reserved broth with remaining teaspoon salt. Simmer, covered, stirring occasionally, until potatoes are almost tender, about 10 minutes.
- Add potatoes and onions to chorizo mixture along with 2 cups broth (save remainder for another use). Stir in chicken and simmer 10 minutes. Serve sprinkled with cheese.
- *Available at Latino markets and some specialty foods shops.
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