Preheat oven to 375°F (190°C). Melt 2 tbsp butter in a large oven-safe skillet over medium-high heat. Season chicken with salt and pepper, then cook, stirring occasionally, until golden and cooked through, 6 to 8 minutes. Transfer to a plate.

Blanch broccoli in a pot of salted boiling water until bright green and just tender, 2 to 3 minutes. Drain, rinse with cold water, and shake dry well.

In the same skillet over medium heat, melt remaining 2 tbsp butter. Whisk in flour and cook 1 minute. Slowly whisk in milk and broth, bring to a simmer, and cook until thickened, 3 to 4 minutes. Remove from heat, then stir in 1 cup cheddar, garlic powder, onion powder, 1/2 tsp salt, pepper, and mustard if using.

Fold chicken and broccoli into the sauce. Smooth the top and sprinkle with remaining 1/2 cup cheddar. If using panko, mix it with the 1 tbsp melted butter and scatter on top. If not using an oven-safe skillet, transfer mixture to a greased 9x13-inch baking dish first.

Bake until bubbly and golden on top, 20 to 25 minutes. Let rest 5 minutes, then scoop and serve.