Preheat oven to 350°F. Grease an 8-inch baking dish with butter; set aside. Melt butter in a large skillet over medium-high heat. Add onion and cook, stirring often, until deep golden brown, about 10 minutes. Add mushrooms, salt and pepper and cook until golden brown, 8 to 10 minutes. Carefully add beer and cook, scraping up any browned bits, for 2 minutes more. Remove skillet from heat, stir in broccoli, 1 3/4 cups shredded cheese, sour cream and Neufchâtel until well combined and transfer to prepared dish. Scatter remaining 1/4 cup shredded cheese over the top and bake until bubbly and golden brown, about 35 minutes. Serve immediately.
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