Preheat oven to 400 degrees. In a large straight-sided skillet, simmer 1/2 cup water over medium-high heat, then add broccoli and a pinch of salt. Cover, reduce heat to medium-low, and cook until crisp-tender, about 6 minutes. Drain well, then transfer to a large bowl; set aside. Wipe out skillet.

Pat chicken dry; season with salt and pepper. Melt 2 tablespoons butter over medium-high heat. Add half the chicken and cook, turning once, until browned and cooked through, about 4 minutes. Transfer to a plate. Repeat with remaining chicken; set aside.

3. In same skillet, melt 2 tablespoons butter, then add onion and mushrooms. Cook, stirring occasionally, until tender and starting to brown, 6 to 7 minutes. Add flour and cook, stirring, 30 seconds. Slowly add milk, reduce heat to medium, and cook, stirring constantly, until mixture is thick enough to coat the back of a spoon, about 9 minutes. Add 2 cups cheese, Dijon, Worcestershire, cayenne, and sour cream; stir until combined. Season with salt and pepper. Add cheese sauce to bowl with broccoli, followed by chicken and accumulated juices; toss to combine. Transfer to a 2 1/2-quart baking dish.

In a small bowl, combine 2 tablespoons melted butter, panko, and remaining 1 cup cheese. Sprinkle over broccoli-and-chicken mixture. Bake until bubbly and golden, 20 to 25 minutes. Let cool 15 minutes, then serve over noodles or rice.