Season both sides of the chicken with dry mustard, salt, and pepper. Heat the oil in a large Dutch oven (or large skillet) over medium-high heat. Carefully lower the chicken into the pot and cook for about 3-5 minutes per side to sear (just until they begin to brown slightly). Transfer the still-raw chicken to a plate.
Reduce the heat under the pot to medium. Add chopped onion and optional mushrooms. Cook, stirring constantly for about 1 minute. Add the wine to deglaze the pan, cooking for about 1 minute more.
Add the cream and half of the tarragon. Return the chicken to the pot and adjust the heat so that the cream gently simmers. Cover and cook, turning chicken occasionally, until it's firm and cooked through, about 25-30 minutes.
Meanwhile, cook the egg noodles according to package directions. Drain and toss with butter, season with salt and pepper to taste. Keep warm.
Remove chicken pieces from the pot, raise heat to medium-high and continue to simmer the sauce, uncovered, for a few minutes to thicken. Re-season with remaining tarragon and additional salt and pepper if needed.
Remove about 1 cup of sauce and toss it with the egg noodles. Plate up the noodles, then place the chicken over the noodles and sprinkle with parsley.