Cut chicken into 1/2-inch wide strips and season with salt and pepper. Combine in a medium mixing bowl with 2 tablespoons vegetable oil. Toss to coat.
Heat remaining oil in a large skillet over high heat. Fry chicken strips until golden all over but pink inside, about 1 minute per side. Remove from pan and reserve.
In same pan, add white mushrooms and shallots and cook over moderate heat until soft, about 2 minutes. Add caraway seeds and carefully cook just until aroma is released, about 1 minute. (It's easy to burn the seeds.) Remove from heat. Add Madeira and cook over high heat until reduced by half, and stir in drained soaked mushrooms, stock and cream. Reduce by about half, or until cream has thickened sauce to suit your taste.
Reduce heat, stir in remaining seasonings, and return chicken and juices to pan to reheat. When sauce returns to a boil, remove from heat, and serve. Spoon chicken over a bed of Wide Egg Noodles. Ladle on sauce and serve immediately.