Pound the chicken: Place each breast between plastic wrap and pound to an even 1/4-inch thickness. Season both sides with 1 teaspoon salt and 1/2 teaspoon pepper.
Fill and roll: Spread 1 teaspoon Dijon on each breast. Top with 2 slices ham and a quarter of the Gruyère, leaving a 1/2-inch border. Roll tightly from the narrow end, tucking in the sides. Secure with toothpicks.
Chill: Wrap rolls in plastic and refrigerate 30 minutes if time allows.
Preheat: Heat oven to 400°F. Set a wire rack over a rimmed baking sheet.
Breading station: In three shallow dishes, combine flour with 1/2 teaspoon salt and 1/4 teaspoon pepper; beat eggs with water; toss panko with melted butter and 1/4 cup Parmesan until evenly coated.
Bread: Dredge each roll in flour, dip in egg, then press firmly into the buttered panko, coating the ends well.
Sear: Heat olive oil in a large skillet over medium-high. Sear rolls seam-side down first, about 2 minutes per side, until golden.
Bake: Transfer to the wire rack and bake 20 to 25 minutes, until the chicken reaches 165°F on an instant-read thermometer.
Sauce: Melt 2 tablespoons butter in a small saucepan over medium heat. Whisk in 2 tablespoons flour and cook 1 minute. Gradually whisk in milk and simmer 3 to 4 minutes until thickened. Off heat, stir in Dijon and 1/4 cup Parmesan. Season to taste.
Rest and serve: Rest rolls 5 minutes, remove toothpicks, slice on a diagonal, and serve with the Dijon sauce.