Preparation

• 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 to the pan, and cook until golden brown and cooked through, 5 to 7 minutes per side. Transfer to a plate and 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. Continue cooking, about 1 minute more. Whisking constantly, slowly add the milk and cook until it begins to thicken, about 4 minutes more. Reduce the heat to medium-low and add the cream cheese and parmesan to the pan, whisking until thoroughly combined. Whisk in the salt, black pepper, and paprika. Add the processed cheese, continuing to whisk until completely melted. Add half of the cheddar and whisk until melted. Finally, fold in the broccoli and shredded chicken.
• Melt the remaining 2 tablespoons butter and stir into the crushed crackers until well combined.
• Spread half of the rice evenly over the bottom of the prepared baking dish and cover with half of the broccoli-chicken mixture. Repeat the layers once. Top the casserole evenly with the remaining cheddar and the cracker mixture. Bake, rotating once halfway through, until golden and bubbly, about 30 minutes. Allow to cool slightly before serving.