In a large skillet, heat the butter over medium heat. Add celery and onion and stir until translucent and soft, about 3 minutes.
⅓ cup unsalted butter, ¾ cup diced onion, ¾ cup diced celery
Add garlic and flour, whisking to combine. Continue to whisk as you pour in the evaporated milk and whisk until fully combined. Season with salt and pepper to taste.
2 tablespoons diced garlic, 6 tablespoons all-purpose flour, 24 ounces evaporated milk
Cook the sauce until thickened, about 2-3 minutes.
Pour sauce into slow cooker and add chicken broth and broccoli. Stir to combine. Cook on high for 3 hours or on low for 6 hours.
5 cups low-sodium chicken broth, 5 cups diced broccoli
Turn off the heat and stir in the heavy cream. Stir in all the cheese (in batches) stirring to melt and combine. Season with salt and pepper to taste.
½ cup heavy cream, 14 ounces freshly shredded sharp cheddar cheese, 2 ounces freshly shredded Monterrey jack cheese, Kosher salt and freshly ground black pepper
Serve hot garnished with sour cream, cilantro, and more cheese if desired.
Sour cream, Chopped fresh cilantro