Preheat the oven to 450° F.
Peel and core the apples, leaving them whole. Slice the apples horizontally into even 1/4-inch-thick slices, reassemble the slices to form whole apple shapes, and brush the apples with the lemon juice.
Put the apples in a pie plate, fill the cavities with some of the brown sugar, butter, and rum, and sprinkle the apples with the remaining brown sugar, butter, and rum.
Bake the apples in the middle of the oven, basting frequently, for 25 minutes. Reduce the temperature to 350° F, insert the cinnamon sticks in the cavities of the apples, and bake the apples, basting frequently, for 20 minutes more, or until they are tender and browned well and the juices are thick and syrupy.
Divide the apples among 4 plates, spoon the syrup over them, and garnish the apples with the mint leaves.Serves 4