Using a very sharp knife, carefully slice each chicken breast in half horizontally, then Put the chicken breasts on a cutting board and lay a piece of plastic wrap over them. Pound the chicken breasts with a meat mallet or rolling pin until they are about a quarter-inch thick.
Put the flour in a shallow platter and season with salt and pepper and mix to combine.
In a separate shallow bowl, beat the eggs with 3 tablespoons of water.
Dip each piece of chicken in the egg and then dredge both sides of the chicken in the seasoned flour to coat completely, then shake off the excess.
Heat the 1 tablespoon of butter and 1 tablespoon of the oil over medium-high heat in a large skillet.
Add the coated chicken to the hot pan and fry for 2 minutes on each side until golden, turning once. Remove to a large platter in a single layer to keep warm.
Melt the remaining butter and oil in the same pan and lower the heat to medium low.
Add the shallots and garlic to the pan and cook until softened.
Raise the heat to high. Add the wine to the pan, and mix well to stir in any browned bits that may have stuck to the pan. Bring to a boil and cook down to reduce by half.
Mix in the chicken stock, mustard, and Creme Fraiche, bring to a boil.
Stir in the parsley and return chicken to the pan with any accumulated juices. Lower the heat and cook for another 5 minutes to cook the chicken and allow the sauce to thicken and reduce. Taste for seasoning and add more salt as needed before serving.
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Notes
You're only browning the chicken at the start. It will finish cooking in the pan. The easiest way to tell if the chicken is done is to use an instant-read thermometer. You're looking for an internal temperature of 165°F/74°C.
You can switch the type of mustard, but keep in mind that you may need to do a bit of tweaking as some mustards are pretty strong and spicy.
Chicken thighs also would go great in this dish so if you're a fan of dark meat – have at it!
Be sure to allow the wine to simmer and reduce by half to evaporate the alcohol and enhance the sauce's flavor.
Give the sauce 4-5 minutes once you've returned the browned chicken to the pan. The sauce will thicken and reduce, which is the crowning glory of this dish!
If your sauce gets too thick, you can loosen it with a bit of extra stock or water. Don't use wine as you have time to cook out the alcohol.