Melt butter in a skillet over medium-high heat. Add chicken and onion, and sauté until chicken is browned and onion begins to caramelize, about 5 minutes. (You can also do this right in the crock of many modern slow cookers by using the cooker’s browning function.)
Place browned chicken and onion in a slow cooker and cover with broccoli, potatoes, chicken stock, poultry seasoning, salt, and pepper. Stir to combine.
Set the cooker to LOW, cover, and let cook 6 hours.
Whisk flour into milk, and stir into soup. Re-cover, set the cooker to HIGH, and let cook 1 additional hour.
Turn off heat and add Cheddar cheese, stirring just until melted and combined. Serve immediately.