Whisk together the mustard, 2 teaspoons of oil, balsamic vinegar, sugar, oregano and salt an pepper in a bowl. This will emulsify and be thick and paste like.
Rinse and pat dry the chicken with paper towels. Using a sharp knife, horizontally cut a pocket into each chicken breast, be careful not to cut all the way through.
Brush the chicken inside and out with the balsamic dressing.
Place on work surface and layer each breast with 4 basil leaves, the mozzarella cheese slice and top with sun dried tomato. Secure using toothpick or skewer on the diagonal.
Heat the remaining 2 teaspoons of olive oil in an ovenproof skillet (I use cast iron) over high heat. Carefully add the chicken and sauté for 1-2 minutes on each side, or until golden.
Transfer skillet with chicken to oven and cook for 15 minutes, or until cheese is melted and bubbly and chicken is cooked through. Allow to rest for 2-3 minutes before serving. Garnish with torn basil leaves and serve along sautéed vegetables.
Notes
1. I do not pound the chicken breast thin and slice. Keep hand firmly on top of chicken breast and gently with sharp knife cut pocket horizontally. Be careful not to cut through or cut a hole or cheese will melt through. 2. I cut each slice of mozzarella 1/4 inch thick each, about 4 ounces total.