How to Make It
In a blender, puree the basil with the 1/2 cup of olive oil. Pour the puree into a small bowl and let stand at room temperature for about 20 minutes, or until the oil separates from the solids. Skim 1/4 cup of the basil oil into a clean bowl, leaving the solids behind. Stir in the lemon juice and season with 3/4 teaspoon of salt and 1/4 teaspoon of pepper.
In a medium saucepan, heat 1/2 inch of peanut oil over moderate heat until shimmering. Add half of the shallots and stir well. Reduce the heat to moderately low and fry, stirring once or twice, until the shallots are lightly browned and crisp, about 15 minutes. Remove the shallots to paper towels with a slotted spoon. Repeat with the remaining shallots.
Light a grill or heat a grill pan. Brush the shrimp with olive oil and season with salt and pepper. Grill for about 2 minutes per side, or until opaque throughout.
Cut the apple into matchsticks. In a bowl, toss the tatsoi, cabbage, apple and shallots. Add the dressing and toss well. Arrange the salad on 8 plates, top each serving with 3 grilled shrimp and serve.