Rinse the short-grain white rice under cool running until the runoff is mostly clear. Combine the rice with the water in a medium saucepan, bring to a boil, then cover, reduce to low, and cook until tender, 15β18 minutes. Remove from heat and rest, covered, 10 minutes; fluff and divide among 4 warm bowls, keeping covered.
Bring a large pot of water to a boil. Blanch the soybean sprouts until crisp-tender, 3β4 minutes; drain well. In a medium bowl, mix the sesame oil with the kosher salt; toss half of this mixture with the warm soybean sprouts and set aside.
In the same pot, blanch the spinach until just wilted, 20β30 seconds; drain, rinse under cold running, and squeeze very dry. Toss the spinach with the remaining sesame oilβsalt mixture and half of the garlic.
Set aside 1 teaspoon of the vegetable oil for frying eggs. Heat the remaining vegetable oil in a large skillet over medium-high heat. Add the zucchini and stir-fry until tender-crisp, 2β3 minutes; push to one side. Add the carrot and stir-fry until bright and just tender, 2β3 minutes; push aside. Add the shiitake mushrooms and cook until tender and lightly browned, 3β4 minutes. Transfer all vegetables to plates.
In a bowl, combine the ground beef with the soy sauce, the remaining garlic, the scallions, and the black pepper; mix to season evenly. Return the skillet to medium-high heat and cook the beef, breaking it up, until just cooked through and most liquid evaporates, 3β4 minutes.
Heat a small nonstick skillet over medium heat with the reserved oil. Crack in the eggs and fry sunny-side up until the whites are set and the yolks are still runny, 3β4 minutes.