While the water is coming up to a boil to cook the egg noodles, start the mushrooms. Place a large skillet over medium-high heat with 2 tablespoons of butter and 2 tablespoons EVOO. Add the shallots and cook stirring frequently for 2-3 minutes. Add the sliced mushrooms and continue to cook for 4-5 more minutes, until they are tender and barely golden in color. Season the mushrooms with a little salt, pepper and the ground thyme. Add the flour and continue to cook for 1 minute, whisk in the white wine, then the chicken poaching liquid and the heavy cream. Season with a little nutmeg and bring up to a simmer.
Slice 4 of the poached chicken breasts on an angle, scoot each breast into the creamy mushroom sauce and simmer until the chicken is heated through again. Add more of the poaching liquid if the sauce becomes too thick.
Divide the buttered chive egg noodles between 4 serving plates. Top each serving with sliced chicken breast and top with some of the creamy mushroom sauce.