Prepare all the ingredients by cutting and placing each into separate sections on a baking sheet. Remove the stems from the shiitake mushrooms and slice them into 1/4-inch-thick pieces. Peel and julienne the carrots into thin matchsticks, yielding about 2 cups. Do the same with the zucchini squash, cutting them into slender strips to create about 3 cups. For the radish, peel and slice crosswise into thin rounds, then stack them together and cut into matchsticks, yielding about 4 cups.

Fill a large saucepan about three-quarters full with water and bring it to a rolling boil over medium-high heat. Add 1 teaspoon of kosher salt and blanch the spinach in two batches, stirring occasionally, until wilted, about 2 minutes. Drain immediately, then once cool enough to handle, squeeze out as much excess moisture as possible. Transfer the spinach to a small bowl, finely grate one garlic clove over it, and season with 1/2 teaspoon salt and 1 tablespoon toasted sesame oil. Mix well with your hands until evenly combined.

Using the same saucepan, refill it with water and bring it to a boil. Add 1 teaspoon of salt along with the soybean sprouts. Reduce the heat to a gentle simmer, cover, and cook for 20 minutes. Meanwhile, prepare the remaining vegetables.

In a large nonstick frying pan or cast-iron skillet, heat 1 tablespoon of neutral oil over medium-high heat until shimmering. Add the carrots along with 1/2 teaspoon of salt, stir-frying for 2 to 3 minutes until they become slightly tender and their color brightens. Transfer them back to the baking sheet and wipe the pan clean.

Repeat the process with the zucchini, heating another tablespoon of neutral oil over medium-high heat. Stir-fry with 1/2 teaspoon of salt until just tender, about 2 to 3 minutes. Remove from the pan and return to the baking sheet. Wipe the pan once more before proceeding with the mushrooms.

Increase the heat to high and add 1 tablespoon of neutral oil to the same pan. Toss in the sliced shiitake mushrooms along with 1 tablespoon soy sauce and 1 teaspoon toasted sesame oil. Finely grate one garlic clove directly over the mushrooms, then stir-fry for about 4 to 5 minutes, allowing them to develop a deep, rich sear. Transfer to the baking sheet and wipe the pan clean.

In the same pan, heat another tablespoon of neutral oil over medium-high heat. Add the julienned radish, seasoning with 1/2 teaspoon salt and 1/3 cup water. Stir occasionally as the radish softens, becoming translucent but not browned, about 8 to 10 minutes. As it nears doneness, drizzle in 1 tablespoon toasted sesame oil and stir-fry for an additional minute before transferring back to the baking sheet.

Once the soybean sprouts have finished simmering, drain and place them in a medium bowl. Grate the remaining garlic clove over the sprouts, season with 1/2 teaspoon salt, and drizzle with 1 tablespoon toasted sesame oil. Use your hands to mix well, ensuring even seasoning throughout.

In a small bowl, combine the gochujang, remaining 2 tablespoons toasted sesame oil, mirin, and honey. Whisk until smooth and well blended.

Wipe the frying pan clean and heat the remaining 1 tablespoon neutral oil over medium heat. Swirl the oil to evenly coat the surface, then carefully crack in the eggs, spacing them apart. Lower the heat, cover, and cook undisturbed until the egg whites are set while the yolks remain runny, about 2 1/2 to 3 1/2 minutes.

To assemble, place 1 cup of hot cooked white rice into each serving bowl. Arrange the prepared vegetables in separate sections over the rice, dividing the spinach, mushrooms, carrots, zucchini, sprouts, and radish evenly. Top each bowl with a fried egg, then sprinkle with toasted sesame seeds. Drizzle 1 to 2 tablespoons of the gochujang sauce over the bibimbap, serving any extra sauce on the side. Enjoy immediately.