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Heat 2 tablespoon oil in a large Dutch oven or other heavy pot over medium heat. Add mushrooms and cook, tossing occasionally, until browned and tender, 5–8 minutes. Reduce heat to medium-low. Add shallots and 1 teaspoon thyme, season with sea salt and pepper, and cook, stirring often, until shallots are translucent and softened, about 2 minutes. Add stock and reduce heat to low. Bring to a simmer and cook until some of the stock has evaporated, 5−7 minutes.
Meanwhile, bring a large pot of salted water to a boil and cook pasta for 1 minute.
Using tongs, remove pasta from water and add to mushrooms along with 1 cup pasta cooking liquid. Bring to a simmer, and cook, tossing constantly, until pasta is al dente and liquid is slightly thickened, about 2 minutes. Add cream, and continue to cook, tossing, until pasta is coated, about 1 minute. Remove from heat, add butter, and toss to combine. Season with sea salt and pepper, to taste.
Divide pasta among bowls. Top with more thyme and grated parmesan, if using.