Preheat your grill to medium heat (about 375°F).
Butterfly each chicken breast by cutting it horizontally through the thickest part, stopping just shy of cutting all the way through. Open it up like a book.
Place each breast between two sheets of parchment, and gently pound it to an even 1/4-inch thickness. Be careful not to tear the chicken.
Drizzle the chicken with olive oil and season both sides with salt and pepper.
Lay a slice of Swiss cheese, a slice of ham, and another slice of Swiss cheese on one side of each chicken breast. Roll up the chicken tightly into a log and secure it with toothpicks. If there are any tears in the meat, roll that side inward to keep the cheese contained.
Rub the outside of each roll with olive oil and season with a bit more salt and pepper.
Place the chicken rolls on the grill. Cook for 15 to 20 minutes, turning occasionally, until the chicken is cooked through and the internal temperature reaches 165°F.
Remove the chicken from the grill and let it rest for 5 to 10 minutes. This helps the juices redistribute and makes slicing easier.
While the chicken rests, whisk together the sour cream, 3 tablespoons of water, lemon juice, Dijon mustard, and garlic for the sauce. Season with salt and pepper to taste.
Spread the sauce onto a serving platter.
Carefully remove the toothpicks from the chicken, turning them gently if they stick. Slice each roll into pieces and arrange them over the sauce. Sprinkle with fresh chives and serve immediately.