Heat Olive oil and butter in a Dutch oven over medium heat until very hot.
Season Chicken pieces with salt and pepper.
Add chicken to skillet (as many pieces as can fit without crowding the pan) and sear until golden brown. Chicken will splatter. Adjust flame or heat, you don’t want burned chicken. Do not turn chicken pieces until nicely colored. Flip chicken pieces and sear on other side until golden brown. When all chicken has been browned remove from Dutch oven and set aside.
Lower heat to low and add butter (you cannot go wrong with more butter!) if pan looks dry. Add shallot and onion and cook until softened, followed by minced garlic if using. Add Champagne (Dry white wine can be substituted) and chicken broth. Adjust heat to medium or medium high and simmer the liquid for a few minutes. With a spatula scrape up the bottom of the pan to add all the flavor from the remnants of the cooked chicken.
Add the mushrooms to the sauce and cook until mushrooms look nearly done. Adjust salt and pepper to taste. (You can also cook the mushrooms in a different pan and add the cooked mushrooms to the Dutch oven when done).
Put chicken back in Dutch oven, cover and continue cooking at a simmer for about 10 minutes or until chicken reaches 163 degrees (chicken will reach temperature of 165 through carryover).
Open the pan and mix in the crème fraiche (or ½ and ½ or heavy cream). Cream can be added sooner but make sure you are on simmer at that point. Let simmer for a couple minutes. Put the flour into a small mixing bowl and with a spoon slowly add some liquid from the pan. Mix well to blend flour and liquid. Add to the pan to thicken the sauce. (you can thicken the sauce to your preference but sauce should not be too thick or too liquid). Simmer a few more minutes.
Serve and garnish with Fresh parsley.