Cook rice: Place rice in a small saucepan with 1 cup water and 1⁄2 teaspoon salt and bring to a boil. Cover and cook over low until rice is tender and water is absorbed, about 15 minutes. Remove from heat and let sit for 5 minutes. Fluff with a fork, then cover to keep warm until ready to serve.
Prep ingredients: Wipe shiitake mushrooms with a damp paper towel. Then trim stem ends from the shiitake mushrooms, then thinly slice caps. Peel carrots and thinly slice on an angle. Peel and finely chop 1 large garlic clove. Trim ends from scallions, then thinly slice, keeping white and dark greens separate. In a small bowl, whisk gochujang, vinegar, 11⁄2 teaspoons sugar, and 2 teaspoons water.
Start stir-fry: Heat 1 tablespoon oil in a large nonstick skillet over medium-high. Add carrots and cook, without stirring, until golden brown on one side, 2-3 minutes. Add mushrooms, stir, and sprinkle with salt. Cook, stirring, until mushrooms are golden brown and tender, about 5 minutes.
Finish stir-fry: Add garlic and scallion whites to skillet and cook until fragrant, about 1 minute. Add spinach and cook until just wilted, about 1 minute. Season to taste with a pinch of salt and pepper. Transfer to a bowl and cover to keep warm.
Fry eggs: Heat 2 tablespoons oil in same skillet over high. Carefully crack 2 large eggs into the pan (they may splatter), and sprinkle lightly with salt and pepper. Cook until the edges are light brown and crispy and the whites are just set, about 1 minute. Cover pan and cook 1 minute; edges should be very crispy and yolks still runny.
Finish rice & serve: Stir half of the furikake and half of the scallion greens into rice. Spoon rice into bowls. Top with vegetables, eggs, and some of the gochujang sauce serving the rest at the table. Garnish with remaining furikake and scallions. Enjoy!