Preheat the oven to 350°. Grease a 9-inch-square baking dish.
In a medium saucepan of salted boiling water, blanch the broccoli stems and florets until just tender, about 2 minutes. Drain well.
Season the chicken thighs generously with salt and pepper. In a large enameled cast-iron casserole, melt 1 tablespoon of the butter over moderate heat. Add half of the thighs and cook, turning once, until lightly browned and cooked through, about 12 minutes. Transfer to a plate. Repeat with 1 more tablespoon of the butter and the remaining thighs. Slice the chicken into 3/4-inch-wide strips.
In the casserole, melt the remaining 6 tablespoons of butter over moderate heat, scraping up any browned bits with a wooden spoon. Add the onion, mushrooms, garlic and a pinch each of salt and pepper and cook, stirring occasionally, until the vegetables are softened, about 10 minutes. Sprinkle in the flour and cook, stirring frequently, until incorporated, about 2 minutes. Add the chicken broth and heavy cream, bring to a boil and cook, stirring frequently, until the mushroom sauce is thick enough to coat the back of a spoon, about 3 minutes. Remove from the heat and add 1 cup of the cheese, the mayonnaise, lemon juice and smoked paprika. Stir until the cheese is melted. Add the chicken and broccoli and stir to coat, then transfer the mixture to the prepared baking dish.
In a small bowl, combine the remaining 1 cup of cheese with the crushed potato chips. Sprinkle over the chicken and bake for about 30 minutes, until lightly browned and bubbling. Garnish with chopped parsley and serve with rice. Pass hot sauce at the table.
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