Step 1

Season chicken with salt and pepper. Heat 2 tablespoons oil in a pot over medium-high. Add chicken, skin-sides down, and cook, flipping once, until browned all over, 7 to 9 minutes (do not crowd pan; brown in batches if necessary). Add 2 onion wedges, broth, and 3 cups water.

Step 2

Bring to a boil, then reduce heat to medium and cook at a bare simmer (small bubbles sporadically emerging at surface) until chicken is just cooked through, 20 to 25 minutes. Transfer chicken to a plate. Strain broth and return to pot.

Step 3

Meanwhile, heat a heavy skillet (preferably cast iron) over medium-high. Add dried chiles and cook, turning a few times, until blistered in places and fragrant, 1 to 2 minutes. Transfer to a bowl and add enough hot tap water to cover; let stand 10 minutes.

Step 4

Meanwhile, add remaining 1 tablespoon oil to skillet. Add 1 onion wedge, jalapeño, and garlic; cook, turning a few times, until golden brown in places and beginning to soften, 7 to 9 minutes.

Step 5

Transfer mixture to a blender (do not wipe skillet clean), along with reconstituted dried chiles and 1/2 cup soaking liquid, oregano, cilantro, cumin, lime juice, and 1 teaspoon salt. Purée until smooth. Return mixture to skillet and simmer over medium heat, stirring occasionally, until reduced and darkened slightly, 8 to 10 minutes.

Step 6

Remove and discard skin and bones from chicken; shred meat into bite-size pieces (leave drumsticks whole, if desired). Return chicken to pot with puréed-chile mixture and hominy. Simmer, stirring occasionally, 15 minutes. Season to taste. Chop remaining onion wedge and serve soup with onion, radishes, cabbage, tostadas, cilantro, and lime wedges.