Meanwhile, heat oil in a large skillet over medium heat. Add mushrooms and Brussels sprouts and cook, stirring often, until the mushrooms release their liquid, 8 to 10 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute. Add sherry vinegar, scraping up any brown bits; bring to a boil and cook, stirring, until almost evaporated, 10 seconds
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Whisk milk and flour in a bowl; add to the skillet with salt and pepper. Cook, stirring, until the sauce bubbles and thickens, about 2 minutes. Stir in Asiago until melted. Add the sauce to the pasta; gently toss.
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