Scrambled Eggs with Shiitake Mushrooms and Feta Cheese

In medium bowl, beat eggs, 2 tablespoons oil, ⅛ teaspoon salt, and pepper with fork until no streaks of white remain. Heat 1 teaspoon oil, shallot, thyme, and remaining ⅛ teaspoon salt in 12-inch nonstick skillet over medium heat, stirring occasionally, until shallot is softened and beginning to brown, 2 to 3 minutes. Add mushrooms and water, cover, and cook, stirring frequently, until mushrooms are softened, 5 to 8 minutes. Uncover and continue to cook until moisture has evaporated, 2 to 3 minutes longer. Transfer mushroom mixture to small bowl and set aside. Wipe skillet clean with paper towels.

Heat remaining 2 teaspoons oil in now-empty skillet over medium-high heat until shimmering. Add egg mixture and, using rubber spatula, constantly and firmly scrape along bottom and sides of skillet until eggs begin to clump and spatula just leaves trail on bottom of skillet, 30 to 60 seconds. Reduce heat to low and gently but constantly fold eggs until clumped and just slightly wet, 30 to 60 seconds. Fold in mushroom mixture. Transfer to serving dish, sprinkle with feta, and serve.