Place the thinly sliced cucumber in a small bowl. Add 1 teaspoon rice vinegar, 1/2 tablespoon sesame oil, and a small pinch of sea salt. Toss to coat and set aside.
In another small bowl, combine the shredded carrot with the remaining 1 teaspoon rice vinegar and a small pinch of sea salt. Toss and set aside.
Heat 1 tablespoon sesame oil in a medium skillet over medium-high heat. Add the sliced shiitake mushrooms and cook, stirring occasionally, about 5 minutes or until browned and tender.
Season the cooked mushrooms with 1 teaspoon soy sauce, stir to combine, then remove the mushrooms from the skillet and set aside.
Using the same skillet, reduce heat to medium. Crack both eggs into the skillet and fry sunny-side up until the whites are set and the yolks are cooked to your preference, about 3 minutes.
Divide the 2 cups cooked white rice evenly between two bowls. Arrange the dressed cucumber, the dressed shredded carrot, 1 cup mung bean sprouts (divide between bowls), and the cooked shiitake mushrooms on top of the rice.
Top each bowl with a fried egg. Garnish with chopped scallions, black and white sesame seeds, and gochujang sauce to taste.