Place flour in a mound on a wood surface and make a small well in the middle of the flour. Lightly beat eggs with salt and vinegar. Gradually add egg mixture a little at a time to the flour, combing with a fork to moisten all bits of flour. Knead dough into a cohesive mass. Then continue kneading until smooth and elastic, about 400 strokes. Separate about 1/4 of the dough into a separate ball for frying. Cover each portion of dough with plastic wrap to prevent drying out and let rest for about 20 minutes at room temperature.
Season chicken with salt and pepper and dredge in flour. Heat butter and oil in a large rondeau or saute pan, and brown chicken on all sides. Remove chicken to a utility platter, and add mushrooms to pan and saute until golden brown. Then add shallot and garlic and cook until translucent. Add cherry brandy, then de-glaze pan with wine. Add the chicken stock, bring to a boil and reduce to a simmer. Return the chicken parts to the sauce, add the sachet, cover and cook over low heat until chicken is tender, about 40 minutes.
While chicken is cooking, roll out dough in pasta machine (1/4 at a time) until it is about 1/16th of an inch thick. Cut the reserved portion for frying into thin strips, 2-inches long. Cut the larger portion into ribbons with a pasta roller or pizza cutter. Fry the smaller reserved portion at 350 degrees F until golden brown and place on paper toweling to drain. Cook larger portion in boiling water until al dente, just a few minutes for fresh pasta, and drain.
Gently remove chicken to a platter, strain sauce and return the sauce to the pan. Allow sauce to reduce until it is thick enough to coat the back of a spoon. Finish sauce by removing from heat and whisking in creme fraiche. Season with salt and pepper, as needed.
Place boiled pasta on serving plate and top with chicken. Spoon sauce around chicken and top with fried dough.
Recipe courtesy of Robert Irvine with Alex Schulte, based on the dish of Hubert Martini, Culinary Institute of America