For the pickles: Whisk vinegar, sugar and salt together in medium bowl. Add cucumber and bean sprouts and toss to combine. Gently press on vegetables to submerge. Cover and refrigerate for at least 30 minutes or up to 24 hours.
For the chile sauce: Whisk gochujang, water, oil, and sugar together in small bowl. Cover and set aside.
For the rice: Bring rice, water, and salt to boil in medium saucepan over high heat. Cover, reduce heat to low, and cook for 7 minutes. Remove rice from heat and let sit, covered, until tender, about 15 minutes.
For the vegetables: While the rice cooks, stir together water, scallions, soy sauce, garlic and sugar. Heat 1 teaspoon oil in Dutch oven over high heat until shimmering. Add carrots and stir until coated. Add 1/3 cup scallion mixture and cook, stirring frequently, until carrots are slightly softened and moisture has evaporated, 1 to 2 minutes. Using slotted spoon, transfer carrots to small bowl.
Heat 1 teaspoon oil in now-empty pot until shimmering. Add mushrooms and stir until coated with oil. Add 1/3 cup scallion mixture and cook, stirring frequently, until mushrooms are tender and moisture has evaporated, 3 to 4 minutes. Using slotted spoon, transfer mushrooms to second small bowl.
Heat remaining 1 teaspoon oil in now-empty pot until shimmering. Add spinach and remaining 1/3 cup scallion mixture and stir to coat spinach. Cook, stirring frequently, until spinach is completely wilted but still bright green, 1 to 2 minutes. Using slotted spoon, transfer spinach to third small bowl. Discard any remaining liquid and wipe out pot with paper towel.
For the bibimbap: Heat 2 tablespoons vegetable oil and sesame oil in now-empty pot over high heat until shimmering. Carefully add cooked rice and gently press into even layer. Cook, without stirring, until rice begins to form crust on bottom of pot, about 2 minutes. Using slotted spoon, transfer carrots, spinach, and mushrooms to pot and arrange in piles that cover surface of rice. Reduce heat to low.
While crust forms, heat remaining 2 teaspoons vegetable oil in 10" nonstick skillet over low heat for 5 minutes. Crack eggs into small bowl. Pour eggs into skillet; cover and cook (about 2 minutes for runny yolks, 2 1/2 minutes for soft but set yolks, and 3 minutes for firmly set yolks). Slide eggs onto vegetables in pot.
Drizzle 2 tablespoons chile sauce over eggs. Without disturbing crust, use wooden spoon to stir rice, vegetables, and eggs until combined. Just before serving, scrape large pieces of crust from bottom of pot and stir into rice. Serve in individual bowls, passing pickles and extra chile sauce separately.
For a quick dinner, prepare the pickles, chile sauce, and vegetables a day ahead (warm the vegetables to room temperature in the microwave before adding them to the rice). You can also substitute storebought kimchi for the pickles to save time. The Korean chile paste gochujang is sold in Asian markets and some supermarkets. If you can’t find it, an equal amount of Sriracha can be substituted. But because Sriracha is more watery than gochujang, omit the water from the chile sauce and stir just 1 tablespoon of sauce into the rice in step 9. For a true bibimbap experience, bring the pot to the table before stirring the vegetables into the rice in step 9.