Preheat a charcoal grill for direct grilling over high heat or you can use the broiler, on high.
Whisk together the miso, sake, sugar and 2 tablespoons water until combined.
Brush the cut-side of the eggplant with the oil and season with salt and pepper.
Grill the eggplant, cut-side down, until lightly golden brown, about 4 minutes. Turn the eggplant over, brush with some of the glaze and continue grilling for 2 more minutes. Brush with more glaze, flip over and grill until the cut-side is caramelized, about 4 minutes.
Remove to a platter, cut-side up, and sprinkle with mint.