Preheat the oven to 180 °C (350 °F).
Pat the duck breasts with a paper towel. Using a knife, make shallow incisions in the skin in a grid-like pattern, taking care not to cut into the flesh. Season with salt and pepper.
Preheat an ovenproof skillet over high heat. Place the duck breasts in the skillet, skin side down. Sear for 2 minutes. If necessary, drain excess grease from the pan during cooking to obtain a crispier skin. Turn the duck breasts over and sear for 2 minutes. Finish cooking in the oven for 9 to 11 minutes.
Remove from the oven and transfer to a cutting board. Cover loosely with a sheet of aluminum foil. Let stand for 5 minutes before slicing.
Drain the leftover grease from the pan. Sauté the mini bok choys for 1 to 2 minutes.
Add the ingredients for the sauce to a saucepan and mix. Bring to a boil, then reduce heat to low and simmer until the sauce develops a syrupy consistency.
Divide the sliced duck meat and mini bok choys among the plates and pour the sauce over the duck. Serve with rice, if desired.