For the pickled vegetables: In a large nonreactive bowl, combine the cucumber, cabbage, carrot, and radishes.
In a small saucepan, combine the apple cider vinegar, honey, and salt and bring to a simmer, stirring until the salt has dissolved. Pour the pickling liquid over the vegetables, cover, and refrigerate for 30 minutes.
For the gochujang sauce: In a small bowl, whisk together the gochujang, mirin, rice vinegar, honey, sesame oil, soy sauce, and garlic. Set aside.
For the hot vegetables: In a large sauté pan, bring an inch or so of water to a boil over high heat. Add the spinach and stir just until spinach is bright green and barely wilted, about 1 minute. Drain the spinach and gently squeeze out excess moisture, then transfer the spinach to a bowl, add the soy sauce and 1 teaspoon of the sesame oil, and toss to coat. Set aside.
Return the pan to medium heat (add a little water if the pan is dry) and add the bean sprouts, if desired. As soon as they begin to wilt, about 10 seconds, transfer them to a bowl and set aside.
Wipe the pan dry and return it to medium heat. Add the remaining 2 teaspoons sesame oil and the mushrooms and season lightly with salt. Sauté until the mushrooms are tender and browned, about 3 minutes. Remove from heat and set aside.
In a large nonstick skillet, heat the vegetable oil over medium-low heat. Crack the eggs into the pan and fry until the whites are firm but the yolks are still runny, 2 to 4 minutes each.
To serve, divide the hot cooked rice among four large serving bowls. Top with the spinach, mushrooms, bean sprouts, and pickled vegetables. Place a fried egg atop each serving. Drizzle about 1 teaspoon sesame oil over each bowl, add some gochujang sauce and a few strips of seaweed sheets, and finish with a sprinkle of sesame seeds.