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Mix cucumber, chile, lime juice, half of cabbage (i.e., ¼ of the head), and one-third of scallions in a small bowl. Gently massage with your hands to soften cabbage; season with salt. Set aside.
Heat 2 Tbsp. oil in a medium (heatproof if you plan to finish in oven) nonstick skillet over medium-high. Cook eggplant, tossing occasionally, until browned in spots and mostly tender, 5–6 minutes. Add onion, garlic, and remaining cabbage and cook, tossing often, until vegetables are very tender, 4–6 minutes. Transfer to a plate and let cool until just warm (about 5 minutes should do it); reserve skillet.
Whisk eggs, ginger, tamari, and remaining scallions in a medium bowl. Season with salt. Add cooked vegetables and stir to combine.
Heat remaining 1 Tbsp. oil in reserved skillet over medium-high. Add egg mixture, shaking skillet to distribute evenly. Cook until golden brown underneath but only partially set on top, about 3 minutes. Slide frittata onto a plate, then invert skillet onto plate. Firmly grip pan handle in one hand and hold plate flush against rim of skillet with the other. Quickly and confidently flip skillet and plate over so frittata lands in pan with the browned side facing up. Return to medium-high heat and cook until other side is just set, about 2 minutes more.
Transfer frittata to a plate. Top with cabbage slaw and pepper.