Bring a large pot of salted water to a boil. Add pasta and cook for about 2 minutes less than package indicates. You want it to be even less than al dente. Drain and reserve ½ cup for later.
While the pasta is being prepared, season the chicken with 1 teaspoon each of the salt and pepper.
Heat oil in a large skillet, big enough to hold the finished dish. (Preferably NOT non-stick.)
Add the chicken in a single layer and cook over medium heat, undisturbed to brown for about 4 minutes. Stir and allow to cook another 3 minutes. Remove to a bowl. DO NOT wipe the skillet.
Return the skillet to the heat and cook the shallot and mushrooms until mushrooms have released their liquid, about 7 minutes. Add the garlic and cook 1 minutes. Remove the skillet from the heat and add the Cognac. Return pan to the heat and scrape up all the brown bits until. Continue cooking until Cognac is almost evaporated, about 1 minute.
Add the chicken stock, remaining salt and pepper and thyme. Simmer for about 5 minutes until reduced by half.
Add the chicken and pan juices, pasta, creme fraiche and pasta water to the skillet. Stir until well mixed and heated through, about 2 minutes.
Remove skillet from the heat and stir in the Parmesan cheese.
Divide among 4 bowls and garnish with more cheese, black pepper and chives, if desired.