Bring a large pot of salted water to a boil over high heat. Add the pasta and cook for 2 minutes under the package directions, about 9 minutes. Drain well reserving 2 cups of pasta water.

Heat a large skillet over medium high heat. Add the olive oil, mushrooms and 1/2 teaspoon kosher salt to the pan and cook, stirring occasionally with a wooden spoon until the mushrooms are golden brown, about 8 to 10 minutes. Add the shallots and cook an additional 2 minutes to soften. Crumble in the dried thyme and deglaze with the white wine. Cook stirring often until almost evaporated, about 4 minutes. Sprinkle the flour over the mixture and stir with a wooden spoon to create a paste. Cook for 1 minute stirring often. Whisk in the cream and remaining 1 1/2 teaspoons salt and bring to a boil. Reduce the heat to medium and simmer for 5 minutes. Stir in the chicken and 3/4 cup of reserved pasta water. Add the pasta to the sauce and top the bare pasta with the parmesan cheese. Toss well to coat adding more pasta water as need to maintain the saucy consistency. Stir in the peas to warm through. Serve topped with more parmesan cheese if desired.