In a saucepan mix the mirin, sake, miso and sugar with a whisk and then put over medium hear and heat until simmer. Stirring often to make sure not to burn. Remove and reserve.
Using a sharp knife, slice the eggplant in half, lengthwise. Then score the inside flesh of each half with a knife in a crisscross pattern.
Brush eggplant with vegetable oil.
Put eggplant on a smoker at 225*F (107*C) and let cook for 2 1/2-3 hours.
After the eggplant has been on the smoker for 45 minutes, start brushing it with the miso sauce every 15 minutes.
Brush the olive oil on each of the pieces of bread and gill to toast and reserve for service.
At service a final brush with the sauce and garnish with green onion and sesame seeds.
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