Wash and dry all produce. Peel mince or grate ginger. Trim and thinly slice scallions, separating whites from greens. Using a peeler, shave carrot lengthwise into ribbons, rotating as you go, until you get to the core; discard core. Trim and halve zucchini lengthwise; cut crosswise into 1/4-inch-thick half-moons. Trim and thinly slice mushrooms.

Melt 1 TBSP butter in a small pot over medium-high heat. Add ginger and cook, stirring, until fragrant, 30 seconds. Stir in rice, 11/4 cups water (21/4 cups for 4 servings), and a pinch of salt. Bring to a boil, then cover and reduce to a low simmer. Cook until rice is tender, 15-18 minutes. Keep covered off heat until ready to serve.

While rice cooks, in a small bowl, combine scallion whites, vinegar, and a pinch of salt; set aside to marinate. In a separate small bowl, combine sesame oil, soy sauce, 1 TBSP sugar (2 TBSP for 4 servings), and up to 1 tsp sriracha (save any remaining for serving).

Heat a drizzle of oil in a large, preferably nonstick, pan over medium-high heat. Add carrot and season with salt and pepper. Cook, stirring, until just tender, 3-4 minutes. Transfer to a medium bowl. Add zucchini and another drizzle of oil to pan. Cook, stirring, until tender, 5-6 minutes. Season with salt and pepper. Transfer to bowl with carrot. Add mushrooms and another drizzle of oil to pan. Cook, stirring, until tender, 3-5 minutes. Season with salt and pepper. Turn off heat; transfer to bowl with veggies. Wipe out pan.

Heat a drizzle of oil in pan used for veggies over medium heat. Once hot, crack eggs into pan and cover. (For 4 servings, you may want to cook eggs in batches.) Fry eggs to preference. Season with salt and pepper.

Fluff rice with a fork and season with salt and pepper; divide between bowls. Arrange carrot, zucchini, and mushrooms on top. Top each bowl with a fried egg and pickled scallion whites (draining first). Drizzle with sauce and any remaining sriracha to taste. Sprinkle with scallion greens and serve.