Preheat a large pot over high heat. Add olive oil, swirl to coat, and add onions, poblanos, garlic, and reserved chicken skin. Season with salt and pepper and cook until softened and beginning to turn golden in spots, 5 minutes. Make a well in the center of the pot. Add remaining 1 tablespoon olive oil and add cumin, coriander and paprika. Stir to coat and toast until fragrant, stirring, 1 minute. Stir to combine. Add black beans, corn, and chicken broth. Bring to a simmer. Add chicken meat and heat just until warmed through. Remove chicken skin pieces and discard. Add cilantro and season with salt, pepper, and lime juice to taste.
Place a few chips at the bottom of each serving bowl. Fill with soup and top each portion with jalapeño slices, cheddar, scallions, sour cream, lime wedges, additional cilantro, and additional tortilla chips. Serve.