Toast chiles in a dry small skillet over medium-high heat, turning occasionally and pressing down to help them make contact with the pan, until darkened in color and beginning to soften, about 3 minutes. Transfer to a bowl and cover with 2 cups boiling water; let sit 30 minutes to soften. Drain; remove seeds and stems. Purée chiles and broth in a blender until smooth, about 1 minute.
Once chiles have soaked 10 minutes, heat 1 tablespoon oil in a large heavy pot over medium-high. Season beef with salt and pepper and cook, stirring occasionally, until browned all over, about 5 minutes. At this point, quite a bit of liquid will have accumulated; continue to cook until it has evaporated, about 3 minutes. Transfer beef to a plate.
Add 1 tablespoon oil to pot, then white onion and garlic; season with salt and pepper. Cook, stirring often, until softened, 6–8 minutes; onion juices will help loosen anything left on bottom of pot. Add cumin and oregano and cook, stirring, 1 minute.
Return beef to pot and add beer. Bring to a boil, reduce heat, and simmer, stirring occasionally, until almost completely reduced, about 4 minutes. Add chile purée, season with salt and pepper, and bring to a boil. Reduce heat and simmer until liquid is slightly thickened and meat is tender, 20–25 minutes. Add squash; cover and cook until tender, 10–15 minutes. Stir in lime juice.
Meanwhile, preheat oven to 350°. Toss pumpkin seeds and remaining 1 teaspoon oil on a rimmed baking sheet; season with salt. Bake, stirring occasionally, until golden brown, 8–10 minutes. Let cool.
Serve chili topped with sour cream, avocado, red onion, and toasted pumpkin seeds.
Recipe Notes
Note: Chili can be made 3 days ahead. Let cool; cover and chill, or freeze up to 1 month.
Toast mustard and coriander seeds in a dry medium saucepan over medium heat, tossing often, until mustard seeds begin to pop, about 3 minutes. Add vinegar, salt, and sugar and bring to a boil. Reduce heat and simmer, stirring often, until salt and sugar are dissolved, about 4 minutes. Remove from heat; stir in dill and 4 cups water. Let cool.
Chicken
Place chicken and brine in a large resealable plastic bag; chill 3 hours. Remove chicken from brine, scraping off seeds, cover, and chill until ready to fry.
Fit a large pot with thermometer and pour in oil to measure 2”. Heat over medium-high heat until thermometer registers 350°.
Meanwhile, place buttermilk in a large bowl. Place flour in another large bowl; season with kosher salt. Working in batches, coat chicken in buttermilk, then dredge in flour, dipping your fingers in buttermilk as you pack flour on to help create moistened, shaggy bits (the makings of a super-crisp crust); transfer to a baking sheet.
Working in batches and returning oil to 350° between batches, fry chicken, turning occasionally, until skin is deep golden brown and crisp and chicken is cooked through, 6–8 minutes. Transfer to a wire rack set inside a baking sheet.
Drizzle chicken with honey; sprinkle with Tuxedo Blend Sesame seeds. Serve with hot sauce alongside.
Make the spicy chicken topping:
Fill a large pot with 8 cups water; stir in 1/4 cup salt. Add chicken and simmer over medium-low heat until just cooked through, about 45 minutes. Drain, then chop or tear chicken into bite-size chunks. Discard bones.
Meanwhile, purée tomatoes, chiles, adobo sauce, pepper, cinnamon, and cloves in a blender until smooth.
Cook onion in a large skillet over medium-high heat, stirring occasionally, until sweet and charred (you want more char than "caramelization"), 8–10 minutes. Add garlic and cook, stirring occasionally, until golden brown and fragrant, about 3 minutes longer.
Add tomato mixture to skillet (it will splatter), reduce heat, and simmer, partially covered, about 30 minutes. Add chicken and chorizo, if using, and cook, stirring occasionally, until chicken is well-coated and moist and liquid is reduced and thickened, about 10 minutes more. Season with salt, if needed.
Assemble the tostadas:
If frying tortillas, fit a large skillet with candy or deep-fry thermometer; pour in oil to measure 1". Heat over medium-high until thermometer registers 375°F. Fry tortillas in batches until crispy, turning every 30 seconds, about 2 minutes total. Drain on paper towel-lined baking sheets and season with salt.
Spread 1 Tbsp. Creamy Black Bean Spread on a tostada shell. Top with 1/4 cup spicy chicken topping, then lettuce, onion, crema, queso fresco, and cilantro. Repeat with remaining tostada shells. Serve at room temperature.
Do Ahead
Spicy chicken topping can be chilled for up to 3 days. Reheat and let cool to room temperature before using.
