In a small bowl, soak the shiitake mushrooms in very hot tap water until softened, 10 to 15 minutes. Remove the mushrooms, discarding the soaking liquid. Discard the stems and thinly slice the mushroom caps.
While the mushrooms soak, prepare the remaining frittata ingredients.

• Scrub or peel the potatoes. Cut the potatoes in half lengthwise, then crosswise into ¼-inch-thick half-moons.
• Trim the root ends from the scallions; thinly slice the scallions, keeping the white and green parts separate.
• Crack the eggs into a medium bowl and season with salt and pepper. Using a whisk or fork, lightly beat until just blended.

In a medium frying pan over medium-high heat, warm 2 tablespoons oil until hot but not smoking. Add the potatoes, season with salt and pepper, and cook, stirring occasionally, until starting to brown and soften, 6 to 8 minutes. Add the mushrooms and the white parts of the scallions and cook, stirring occasionally, until the scallions start to soften, 2 to 3 minutes. Working in batches if needed, stir in the spinach, season lightly with salt and pepper, and cook until just wilted, 1 to 2 minutes. Continue to cook until the pan is dry and any moisture has evaporated, 1 to 2 minutes.
Spread the vegetables evenly over the bottom of the pan. Pour the eggs on top. Reduce the heat to medium-low, cover, and cook, without stirring, until the eggs are set, 3 to 5 minutes. Remove from the heat and let cool slightly, then cut into wedges.
While the frittata cooks and cools, prepare the kimchi slaw.

• Coarsely chop the kimchi.
• Peel the apple, if desired; cut the apple into quarters and cut away the core. Thinly slice the apple.

In a medium bowl, combine the kimchi, apple, green parts of the scallions, mung bean sprouts, sesame oil, rice vinegar, and black sesame seeds and toss to coat. Season to taste with salt and pepper.

Transfer the frittata wedges to individual plates and serve with the kimchi slaw.