In a frying pan, heat your oil, but don't let it get so hot it smokes. Using a whisk, whisk in the flour a little at a time so that the flour and oil mix well and start to bubble, and become golden in color and fragrant.
Slowly pour in the milk in a steady slow stream, whisking the whole time until a cream is formed. Follow with the chicken or vegetable broth until fully incorporated and then remove from heat.
Pour the mixture into your slow cooker and add the thawed broccoli. We use the frozen broccoli because it seems to hold up better in a slow cooker than the fresh variety. But you can certainly use fresh broccoli florets as well. Add salt to taste, black pepper and garlic, stir to combine.
Cook on low for 4-6 hours, or until the broccoli is completely cooked. Stir in the cheese, 1 cup at a time, allowing it to melt completely before adding the next cup of cheese.
Puree the soup in the blender in batches, filling halfway each time. A hand blender (immersion blender) will also work. Be sure to scrape any of the remaining cheese from the bottom of the slow cooker into the soup to blend well. Blend as little or as much as you like until desired consistency is achieved, i.e. some prefer a smoother and some prefer a chunkier soup