Heat oil in a large oven-safe skillet over medium-high heat. Arrange the stuffed breasts, top-sides down, in the pan; cook until golden, about 2 minutes. Carefully flip the chicken; transfer the pan to the oven. Bake until an instant-read thermometer inserted in the thickest portion of the chicken registers 165 degrees F, 20 to 25 minutes. Drizzle the chicken evenly with lemon juice. Remove the wooden picks from the chicken before serving.