Melt butter in small saucepan over medium heat. Add onion and saute until glossy. Transfer this to your crock-pot Add flour and mix until smooth. Add chicken broth, salt, celery, carrot and broccoli. Cook on low for 6-7 hours, or high for 3-4 hours. Add cheese cubes, stirring until melted. Add half and half and stir until blended. Cover and continue to cook for 30 minutes on low. Serve and enjoy!