Preheat oven to 400°F and position the rack in the top third.
Whisk sauce ingredients together in a small bowl and set aside.
Slice chicken breasts in half lengthwise to create four thin cutlets and season with salt and pepper.
Heat oil and butter in an oven-safe skillet over medium-high heat for several minutes.
Cook chicken for 4 to 5 minutes per side until golden brown.
Whisk sauce again, pour into the skillet around the chicken, and simmer for 30 seconds. Remove from heat, spoon sauce over the chicken, and bake for 4 to 5 minutes until the internal temperature reaches 165°F and the sauce thickens, taking care not to burn it.
Serve immediately, topped with extra sauce and chopped scallions.