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In a large pot of boiling salted water, cook pasta according to package instructions; drain well.
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Heat olive oil in a large skillet over medium heat. Add mushrooms and shallot, and cook, stirring frequently, until tender, about 3-5 minutes. Stir in garlic and thyme until fragrant, about 1 minute.
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Whisk in flour until lightly browned, about 1 minute. Gradually whisk in vegetable broth, and cook, whisking constantly, until incorporated, about 1-2 minutes.
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Stir in half and half and Parmesan cheese and cook, whisking constantly, until slightly thickened, about 2 minutes; season with salt and pepper to taste.
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Stir in orzo and spinach until the spinach has wilted, about 3 minutes.
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Serve immediately.