Set the oven to 375 degrees. Using butter, grease a 2-quart baking dish. Melt 4 tablespoons of butter over medium-high heat in a medium saucepan. Cook the onion and garlic for about 5 minutes, stirring periodically, until soft. Add the flour and stir continuously for one minute before adding the mustard.
Pour milk in gently, followed by broth as you stir. Stirring continuously, bring to a simmer, and cook for 7 minutes or until thickened.
Whisk in cheese, sour cream, and cayenne pepper until thoroughly blended over low heat. Get rid of the heat. Stir in the broccoli and turkey after liberally seasoning with salt and pepper.
Salt and pepper the turkey mixture after spreading it on the bottom of the preheated pan.
For about 40 minutes, roast the dish with foil covered until the broccoli is cooked.
Stir crackers into remaining 2 tablespoons of butter as it melts.
Removing the foil, add the cracker mixture on top. Bake for another 25 minutes or until cheese is bubbling and brown.