In a small bowl, mix together the mustard and mascarpone. Set aside.
Preheat the broiler to high.
Heat a large ovenproof skillet over medium-high heat. Add the oil and continue to heat for an additional 1 minute. Sprinkle the chicken breasts evenly with 1 teaspoon salt. Place the chicken in the skillet and cook on the first side until golden brown, about 4 minutes. Flip the chicken and continue to cook until just cooked through, about 2 minutes more. Remove from the heat.
Lay 1/2 slice of cheese on each chicken breast. Lay 2 slices of ham on top of the cheese on each breast. Divide the mustard sauce evenly over the ham and top with the remaining cheese slices. Broil until the cheese is melted, golden brown and bubbly, about 2 minutes.