FOR THE CHICKEN: Pat chicken dry with paper towels and season with salt and pepper. Heat 1 tablespoon oil in large Dutch oven over medium-high heat until shimmering. Add half of chicken and brown on both sides, 3 to 4 minutes per side. Transfer to large plate. Repeat with remaining chicken.
Reduce heat to medium, add remaining 1 tablespoon oil to now-empty pot, and heat until shimmering. Add onion and cook, stirring frequently, until browned, about 5 minutes. Add garlic, cumin and cinnamon; cook until fragrant, about 1 minute. Add tomatoes, broth, chipotle with adobo sauce and brown sugar; bring to boil, scraping up any browned bits.
Return chicken to pot, reduce heat to medium-low, cover and simmer until meat registers 195 degrees, 15 to 20 minutes, flipping chicken after 5 minutes. Transfer chicken to cutting board.
Transfer cooking liquid to blender and process until smooth, 15 to 30 seconds. Return sauce to pot. When cool enough to handle, use two forks to shred chicken into bite-size pieces. Return chicken to pot with sauce. Cook over medium heat, stirring frequently, until sauce is thickened and clings to chicken, about 10 minutes. Stir in lime zest and juice. Season with salt and pepper to taste.
THE REST: Spoon chicken into center of each warm tortilla and serve, passing avocado, Cotija, green onions, cilantro and lime wedges separately.