Serve with hot sauce and crusty bread.
Melt butter in large saucepan over medium-high heat. Add onion, salt, and pepper and cook, stirring occasionally, until softened, about 4 minutes. Add flour and cook, stirring to coat onion, about 1 minute. Gradually whisk in broth. Add chicken and broccoli and bring to simmer. Cover; reduce heat to medium-low; and simmer until chicken registers 160 degrees, about 10 minutes.
Transfer chicken to plate and shred into bite-size pieces using 2 forks. Stir cream into soup and return to simmer over medium heat. Whisk in cheddar and American cheese until melted, about 2 minutes. Stir in chicken. Season with salt and pepper to taste. Serve.
