Heat a medium sauce pot over medium heat. Add the butter and allow to melt. Stir in the flour, whisking thoroughly and cooking for a few seconds to develop the flavor. Slowly stream in the milk and whisk until the flour and butter have fully dissolved into the milk. Cook until thickened. Remove from heat.
In the bowl of a slow cooker, combine the chicken (or vegetable) broth, onion, garlic, broccoli, cream cheese, seasonings and thickened milk until evenly combined. Cover and cook on low for about 6 to 8 hours or on high for 3 to 4 hours. Before serving, stir in 2 cups of cheddar cheese until melted. Serve with a sprinkling more of cheese if desired. Enjoy!