Cut off and discard the tops and bottoms of the eggplants. Peel the eggplants.
Place the peeled eggplants in a microwaveable bowl and cover with cling wrap. Microwave on HIGH for one minute (see notes after the recipe).
Place the eggplants on a cutting board and cut each horizontally into halves.
Butterfly the eggplants, kabayaki-style. Slit each eggplant half down on the middle but without cutting all the way through. Open up the eggplant and make a similar slit along the left and right portions.
Butterfly the rest of the eggplant halves.
Pour the soy sauce, mirin and sake (and sugar, if using) into a shallow pan. Set on the stove over medium heat and bring to a simmer.
Arrange the butterflied eggplants in the pan in a single layer, cut side down.
Cook the eggplants until softened but not mushy, and the sauce has been soaked up.
Serve the kabayaki-style eggplant braised in teriyaki sauce over hot rice.