Insert a paring knife into the thickest part of each chicken breast to make a 3-inch-long pocket down the side, cutting in about halfway. Rub the insides of the pockets with 1 tablespoon of the mustard for all 4 breasts. Stuff each breast with 2 half slices of ham and 1 slice of provolone, folding them to fit. Rub both sides of the chicken breasts with the olive oil and sprinkle with the salt and pepper.
Heat a nonstick skillet over medium-high heat until very hot, 3 to 5 minutes. Add the chicken breasts and cook until golden brown and just cooked through, 6 to 8 minutes per side depending on the thickness of the chicken; reduce the heat to medium if the chicken becomes too dark towards the end of the cooking time. Cover the skillet with a lid or a piece of aluminum foil during the last 3 minutes of cooking time to finish cooking the chicken through to an internal temperature of 160 degrees F. Check the temperature at the thickest part of the breast without stuffing.
Transfer the chicken to 4 plates or a cutting board and add the chicken broth and remaining 1 tablespoon mustard to the skillet. Cook, whisking, over medium-high heat until the sauce has reduced by about half, 3 to 4 minutes. Remove from the heat and swirl in the butter and chopped herbs.
Depending on the size of the breasts, either put 1chicken breast on each of 4 plates or slice all the breasts and divide among 6 plates. Spoon the sauce over the chicken and serve with Crispy Sliced Roasted Potatoes on the side.
Place a rimmed baking sheet in the center rack of the oven and preheat the oven to 425 degrees F.
Thinly slice the potatoes about 1/8-inch thick and toss with the oil, salt and pepper. Carefully spread the slices on the hot baking sheet in a single layer and roast until golden and crispy, turning with a metal spatula occasionally and spreading them out to cook evenly, 30 to 35 minutes.