In a large pot of boiling salted water, cook the pasta until al dente. Drain and return to the pot. Cover to keep warm.
Meanwhile, in a large saute pan, heat the oil over medium-high heat. Add the mushrooms and cook, stirring, until soft, 3 to 4 minutes. Add the shallots, garlic, thyme, salt, and pepper, and cook, stirring, until fragrant, 2 minutes. Add the cream, increase the heat to high, and bring to a boil. Reduce the heat and simmer until the sauce thickens enough to coat the back of a spoon, about 5 minutes. Add the Parmesan and adjust the seasoning, to taste.
Add the pasta, stir to coat with the sauce, and cook until the pasta is heated through, 1 minute. Remove from the heat and divide among 4 serving bowls or plates. Sprinkle each serving with chives and serve immediately.