Step 1Heat the oven to 350 degrees. Fill a large pot with water, add 1 teaspoon of salt and bring to a boil. Bring a pot of water for the green beans to boil as well.
Step 2Meanwhile, combine the butter and fines herbes in a bowl; set aside. Place 1 chicken breast at a time in a resealable plastic bag and pound it to flatten slightly. Place a chicken breast on a work surface and place 1 tablespoon of the butter mixture in the center. Put 1/4 of the Gruyere cheese on top of the butter. Tuck the sides in the long way, then roll up the breast from the short end and insert a wooden skewer to hold it secure. Finish with the remaining chicken breasts. Save the remainder of the butter mixture for the noodles.
Step 3Heat 3 tablespoons of oil in an ovenproof 12-inch skillet over medium heat. Place the 4 breasts seam side down in the skillet and cook until golden brown, 4 to 5 minutes, then turn them and cook until the other side is golden brown, another 4 minutes. Place the skillet in the oven and cook the chicken until the internal temperature reaches 165 degrees, about 7 to 10 minutes, being careful to insert the thermometer into the chicken and not the hot butter center.
Step 4When the chicken goes into the oven, add the noodles to the boiling water and cook them according to package directions. Add the green beans to the second pot and cook until tender-crisp, 6 to 7 minutes. Drain and toss them with the remaining 2 tablespoons of olive oil. Keep them warm.
Step 5Drain the noodles, stir in the chives and the rest of the butter mixture, then arrange the noodles in the center of each of 4 plates. Remove the skewers from the chicken and slice each breast into 4 to 5 pieces. Divide the pieces among the plates, placing them on top of the noodles, and serve with the green beans.