Season chicken breasts with salt and pepper. Roll in flour and shake off the excess. In a heavy skillet, heat butter and brown lightly the breasts on both sides. Add the chicken stock, cover and simmer for about 10 minutes.
While chicken is simmering, prepare the Cilantro Sauce. In a food processor fitted with the steel blade add 2 cups of cilantro, the pine nuts (or almonds or the mix of both), Parmesan cheese and garlic and blend into a paste. Add softened butter and then slowly pour olive oil through the feed tube until mixture is the consistency of heavy cream. Set aside.
Cook fettuccine according to package directions. Drain and place on a large platter. Toss with butter. Toss in the Cilantro Sauce, reserving some to use as a spread for the chicken.
Place the chicken on a separate platter and sprinkle the remaining fresh cilantro on top.
Serve the remaining Cilantro Sauce on the side. Pass grated Parmesan at the table.