Tips:To clean leeks: Trim and discard coarse green tops. Split leeks lengthwise with a sharp knife, beginning about 1 inch from the root end and cutting toward the green end. Leave root end attached. Swish leeks repeatedly in a basin of cold water to remove grit. Alternatively, trim roots and ragged tops. Slice leeks and place in plenty of water, then drain. Repeat a few times. The slices do not absorb water or lose flavor and the process is faster. Wrap and freeze the chicken tenders. When you have gathered enough, use them in a stir-fry--they are the perfect size. Remove the tough membrane with a sharp knife before cooking.
Trim visible fat from chicken breasts. Remove the tenders (the long thin flaps); reserve for another use (see Tip). Place trimmed chicken breasts between 2 pieces of plastic wrap. Pound with a rolling pin, meat mallet or heavy skillet until flattened to an even thickness, about 1/2 inch. Combine flour, salt and pepper in a shallow glass dish. Dredge chicken in seasoned flour, shaking off excess. (Discard any leftover flour.)
Heat oil in a large nonstick skillet over medium-high heat. Add the chicken and cook until well browned and no longer pink in the center, 4 to 5 minutes per side. Transfer to a plate, cover and keep warm. dd mushrooms and leek to the pan; cook over medium heat, stirring often, until most of the moisture has evaporated and the mushrooms are beginning to brown, 4 to 6 minutes. Add wine (or vermouth) and deglaze, scraping up any browned bits, for about 1 minute. Mix broth and cornstarch in a small bowl. Add to the pan and cook until thickened, 2 to 3 minutes. Stir in sour cream and mustard; cook until heated through, about 1 minute. Season with salt and pepper. Spoon over the chicken and sprinkle with parsley.