Preheat the oven to 350°F. Butter a 13-by-9-inch baking dish and set aside.

Bring a large pot of water to a boil. Add the broccoli and cook until crisp-tender, about 2 minutes. Drain, rinse with cold water, and set aside.

Heat a large high-sided skillet over medium-high. Add 2 tablespoons of the butter and the oil to the pan. When the butter has melted and foamed and the oil is shimmering, add the chicken and cook until golden brown and cooked through, 5 to 7 minutes per side. Transfer to a plate, set aside to cool slightly before shredding. Wipe out the pan and return it to the stove.

Add 2 tablespoons of the butter to the pan set over medium-high heat, and melt. Add the onion and garlic, and cook, stirring occasionally, until softened and translucent, 3 to 4 minutes. Add the flour, dry mustard, and cayenne to the pan and stir to combine.