Preheat oven to 350° F with a rack in the center.
Melt one tablespoon of the butter in a large, oven-safe nonstick frying pan set over medium-high heat.
Add the sliced garlic scapes and cook, stirring once or twice, for one minute.
Add the sliced shiitakes and ¼ teaspoon of the salt and cook, stirring occasionally, until softened, about 5 minutes.
Scrape skillet contents into a bowl and set aside.
Melt another tablespoon of the butter in the same frying pan. Add the grated potatoes in a single layer and sprinkle with remaining salt. Leave them alone for a few minutes to develop a bit of browning on the bottom, then stir as needed to cook through without sticking. Add potatoes to bowl with scapes and shiitakes.
Crack the eggs into a separate bowl, pour in cream, grind in the black pepper, and scramble.
Melt the final tablespoon of butter in the frying pan and pour in egg mixture. Sprinkle in capers.
Return potatoes, scapes, and shiitakes to frying pan, distributing it all evenly around the pan without disturbing the bottom of the eggs.
Sprinkle cheese evenly overtop.
Cook for a minute on the stovetop, then transfer to oven.
Bake until eggs are set, 15-20 minutes. Cool slightly before cutting into wedges to serve.