Pat chicken dry with paper towels. Heat 2 tablespoons oil in Dutch oven over medium-high heat until just smoking. Brown chicken on both sides, 8 to 10 minutes; transfer to plate.
Add broccoli florets to fat left in pot and cook until tender and just beginning to brown, about 4 minutes. Transfer 1 cup broccoli florets to bowl and set aside for serving. Stir broccoli stalks, garlic, mustard, salt, and cayenne into pot and cook until fragrant, about 1 minute. Stir in broth, scraping up browned bits. Return chicken and any accumulated juices to pot and bring to simmer. Reduce heat to low, cover, and cook until chicken registers 165 degrees, about 10 minutes.
Transfer chicken to cutting board, let cool slightly, then shred into bite-size pieces using 2 forks; discard skin and bones.
Working in 2 batches, process soup, spinach, cheddar, Parmesan, and remaining ¼ cup oil in blender until smooth, about 30 seconds for each batch. Return soup and chicken to now-empty pot and bring to brief simmer over medium-low heat. Season with salt and pepper to taste. Top individual portions evenly with chives and reserved broccoli before serving.