Step 1

Season chicken generously with salt. Heat oil in a large Dutch oven over medium-high. Working in batches and adding more oil between batches if needed, arrange chicken, skin side down, in pot in a single layer. Cook, undisturbed, until skin is crisp and golden brown, 5–7 minutes. Turn over; cook until other side is browned, 5–7 minutes. Transfer to a plate.

Step 2

Reduce heat to medium and combine onion and garlic in same pot; season with salt. Cook, stirring occasionally, until onion is softened and starting to brown, about 5 minutes. Add cinnamon stick and cumin seeds and cook, stirring, until fragrant, about 30 seconds. Add tomatoes and their juices, crushing with your hands as you go. Cook, stirring occasionally and scraping up any browned bits, until juices are almost completely reduced, 5–7 minutes.

Step 3

Add chiles and 2 cups broth to pot, then chicken (it should be mostly submerged; add remaining broth if needed). Make sure chiles are under chicken so they can fully hydrate. Increase heat to medium-high; bring to a simmer. Reduce heat to low, partially cover pot, and simmer 45 minutes. Turn chicken over, scraping up browned bits. Partially cover; cook, scraping up browned bits every 15 minutes, until meat shreds easily and braising liquid is reduced by half to three fourths, 45–60 minutes. Transfer chicken to a plate; let cool slightly. Remove skin, then shred meat into a medium bowl; discard skin and bones.

Step 4

Fish out cinnamon stick; discard. Transfer onion, garlic, and chiles to a blender; add ½ cup braising liquid. Blend, adding more braising liquid or broth if needed until smooth (sauce should be the consistency of a loose ketchup); season with salt. Add enough sauce to chicken to coat (save extra sauce for another meal).

Step 5

Spoon chile chicken on tostadas and top with lettuce, red onion, cilantro, and queso fresco. Serve with lime wedges.Do ahead: Chile chicken can be made 4 days ahead. Let cool; cover and chill.