New Mexican or Anaheim chiles can be substituted for the guajillo chiles. If queso fresco is unavailable, you can substitute farmer's cheese or a mild feta. When baking the tortillas, stir them well to promote even browning.
1. Adjust oven racks to upper-middle and lower-middle positions and heat oven to 425 degrees. Divide tortillas evenly between 2 rimmed baking sheets and drizzle each with 2 tablespoons oil and 1/4 teaspoon salt. Toss until tortillas are evenly coated with oil. Bake until golden brown and crisp, 15 to 20 minutes, stirring chips and switching and rotating sheets halfway through baking.
2. Toast guajillos in Dutch oven over medium heat until fragrant and slightly darkened, about 5 minutes. Transfer to blender and process until finely ground, 60 to 90 seconds, scraping down sides of blender jar as needed.
3. Add tomatoes and their juice, 3/4 cup onion, poblano, jalapeño, cilantro sprigs, garlic, and 3/4 teaspoon salt to guajillos and process until very smooth, 60 to 90 seconds. Transfer sauce to now-empty Dutch oven and stir in broth. Bring sauce to boil over medium-high heat. Add chicken breasts; reduce heat to low and simmer, uncovered, until chicken registers 160 degrees, 15 to 20 minutes, flipping halfway through cooking.
4. Using tongs, transfer chicken to large plate. Increase heat to medium and continue to simmer sauce until thickened and reduced to about 4 1/2 cups, about 5 minutes longer. While sauce simmers, shred chicken into bite-size pieces using 2 forks. Return chicken to sauce and cook until warmed through, about 2 minutes.
5. Add chips to pot and toss to coat. Remove from heat and season with salt to taste. Cover and let stand for 2 to 5 minutes, depending on how soft you like your chips.
6. Transfer chilaquiles to serving dish and top with queso fresco, avocado, radishes, remaining 1/4 cup onion, and chopped cilantro. Serve with sour cream and lime wedges.