How to Make It
In a small saucepan, combine the soy sauce with the mirin, sugar and soju; bring to a boil. Simmer over moderate heat, stirring, until reduced to 1/4 cup, about 12 minutes. Let cool completely, then whisk in the mayonnaise and lime juice and season with salt.
Light a grill or preheat a grill pan. Brush the green tomatoes with oil and season with salt. Grill over high heat, turning once, until lightly charred, 4 minutes. Transfer to a plate to cool, then cut into quarters.
In a perforated grill pan, toss the whole grape tomatoes with the 1 tablespoon of oil and season with salt. Grill over high heat, tossing occasionally, until lightly charred, about 2 minutes. Let cool slightly.
On a platter, toss the mozzarella with all of the tomatoes. Drizzle 1/2 cup of the unagi sauce over the salad and scatter the basil and mint on top. Serve, passing the remaining sauce at the table.