Peel and finely chop the shallots. Slice mushrooms.
In a large dutch oven, add 2T Butter. Melt and then add the Shallots. Saute for 5 minutes.
Melt 2T more Butter in with the Shallots. Add the sliced Mushrooms. Sauté for 10 minutes.
Remove the Shallot & Mushroom mix from the pan and set aside.
Melt 2T Duck Fat in the dutch oven. Add the Chicken, skin side down. Sear skin till golden brown. Flip Chicken and sear other side. Continue flipping the chicken for about 20-25 minutes, until it is mostly cooked. Remove from pan and set aside.
Deglaze pan with the Champagne, scraping any bits of cooked on Chicken with a wooden spoon. Simmer for about 6-8 minutes, or until about a third of the Champagne has been evaporated. Remove from heat once proper amount has been evaporated.
While Champagne is cooking, in a separate pot, start the Velouté sauce by preparing a Roux. Melt Butter in pot. Once melted, add Flour. Whisk until a thick paste is formed, becoming slightly beige in color.
Add Chicken Stock - whisking continuously while adding to the pot. Continue to whisk/stir until slightly thickened and the Velouté sauce is formed - about 10 to 15 minutes.
Place Champagne back on the heat, and add the Velouté sauce. Whisk to combine. Add Sour Cream and whisk till combined.
Add the Mushroom & Shallot mixture back in. Stir and simmer for 10 minutes to thicken a bit. *At this point, taste the sauce and season with Salt/Pepper if needed.
Add Chicken back in by nestling each piece in the sauce. Add Sprigs of Thyme. Cook, covered, for 15-20 minutes to finish cooking chicken.
Remove Chicken to plates and top with Champagne Sauce. Enjoy.*Note - One serving option is to nestle the Chicken into bed of rice and ladle the Champagne Sauce over. Very tasty!