Combine 2 cups corn kernels, eggs and sour cream in a blender; season with salt and pepper and pulse to blend until smooth. Transfer to a bowl with remaining corn kernels, scallion whites, cornmeal and flour, and stir just until combined.
Heat ¼ cup olive oil in a large skillet over medium-high heat. Working in two batches, spoon the corn batter into the skillet and pat to flatten into small cakes (about 1/4 cup per cake). Cook, flipping once, until golden on both sides, about 6 minutes per batch.
While cakes cook, stir together remaining 1 tablespoon olive oil and vinegar; add crab, tomatoes and 2 tablespoons scallion greens to bowl and season with salt and pepper. When all the cakes are cooked, toss crab and tomatoes with basil and vinaigrette. Serve cakes topped with crab salad.
Before her diagnosis, Caroline wrote a book on cakes called Cake Magic!. She started developing a birthday cake using her gluten-free mix found in that book. Check out other recipes she’s developing for her new life—and the stories behind them—on her blog, The Wright Recipes. Her next book, Catalan Food, written with chef Daniel Olivella, comes out in early September from Clarkson Potter.