Heat the oven to 350°F. Season chicken with salt and pepper and cook in oil in a skillet for 4 to 5 minutes per side until it reaches 165°F. Cut into pieces that fit individual rolls.
Mix softened cream cheese with Dijon. Split the connected Hawaiian rolls horizontally and place the bottom slab in a parchment-lined 9-by-13-inch pan.
Spread the cream-cheese mixture over the bread. Layer on half the Swiss, cooked chicken pieces, ham and remaining Swiss, then replace the tops.
Brush with butter, sprinkle with breadcrumbs and cover with foil. Bake for 15 minutes, then uncover for 8 to 10 minutes until the filling reaches 165°F.
Rest for 5 minutes and cut into 12 sliders.