How to Make It
In a small saucepan, heat the 1 tablespoon of oil until shimmering. Add the lemongrass, ginger, garlic and chile and cook over moderate heat, stirring, until fragrant and just starting to brown, about 3 minutes. Add the plum and hoisin sauces and cook, stirring, until thickened and reduced to 3/4 cup, about 8 minutes. Scrape the mixture into a blender and puree until nearly smooth.
Meanwhile, light a grill or preheat a grill pan. Brush the eggplant with oil and season with salt and pepper. Grill cut side down over high heat until lightly charred, about 2 minutes. Turn the eggplant and cook until lightly charred and tender, about 2 minutes. Transfer the eggplant to a cutting board and let cool slightly, then cut crosswise into 1 1/2-inch pieces.
In a large skillet, reheat the plum sauce. Add the eggplant and the 2 tablespoons of basil over moderately high heat and cook, tossing, until the eggplant is heated through, about 3 minutes. Transfer to a serving bowl, scatter the cashews and more chopped basil on top and serve.