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Heat 2 Tbsp olive oil in a skillet over medium-high heat. Add asparagus, season with salt and pepper and cook, stirring occasionally, until just tender, about 4 minutes. Transfer to plate.
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Heat butter and remaining 1 tbsp of olive oil in same skillet over medium-high heat. Add mushrooms, season with salt and pepper, and cook, tossing often until tender for about 5 minutes. Add shallot and cook, tossing occasionally, until softened for about 2 minutes. Toss in thyme and cut asparagus. Add heavy cream and parmesan cheese to skillet and cook over low heat.
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Meanwhile, cook pasta in a large pot of boiling salted water (enough salt to taste like ocean watestirring occasionally, until al dente (pasta is slightly hard). Drain reserving one cup of cooking liquid.
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Add pasta and ½ cup cooking liquid to skillet. Cook, tossing and adding more liquid as needed, until sauce coats pasta, about 2 minutes, season with salt and pepper.
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Divide pasta among plates, top with more parmesan cheese.