02While 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.
03Cook 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.
04Peel 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.
05Add 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.
06*Available at Latino markets and some specialty foods shops.
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Not rich — missing servings, total time, or calories.