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Deep, rich miso is a natural partner for the smoky flavor of the grill. For the best caramelization, these eggplant slices are finished off in the oven.
Note: Mirin is a sweet rice-based wine. It can be found in any Japanese or Asian grocery story. If unavailable, you can make a substitue by heating 1 cup of sake with 1 cup of sugar until dissolved.
Combine mirin, sake, miso, and sugar in a small bowl and stir with a fork until a homogenous paste is formed. Set aside.
Rub eggplants on all surfaces with vegetable oil. Heat a grill pan or prepare a gas or charcoal grill to high heat. Grill eggplant, cut-side-down until charred grill marks appear, about 1 1/2 minutes. Rotate 45 degrees and cook until checkered hash marks appear, about 1 1/2 minutes longer. Flip and continue cooking until nearly tender, about 4 minutes longer. Transfer eggplant to a large plate and allow to cool slightly.
Adjust a rack 6-inches from the element and preheat broiler to high heat. Carefully spread the miso glaze on the cut surface of every eggplant. Place on a rimmed baking sheet or foil-lined broiler pan, face up. Broil until completely tender and glaze has begun to caramelize, about 4 minutes. Transfer to a serving platter, sprinkle with sesame seeds and scallions, and serve