Preheat oven to 400°F. Melt 1 Tbsp. butter in a skillet over medium heat. Add onion; sauté until softened, 5 minutes. Add turkey, Worcestershire and thyme, and sauté until turkey is cooked through, 10 minutes.
In a large saucepan over medium heat, melt 2 Tbsp. butter. Stir in flour. Add milk and cook, stirring, until sauce bubbles and thickens, 8 minutes. Remove from heat. Add mustard and 1/2 cup cheese; stir until melted. Add turkey mixture and broccoli. Season with salt and pepper.
Mist a 9-by-13-inch baking dish with nonstick cooking spray, spread mixture in dish and sprinkle with remaining 1 cup cheese. Top evenly with fries. Bake until cheese is bubbling and fries are crisp and golden, 25 to 30 minutes.