Begin by cooking the udon noodles according to the package instructions. Once cooked, drain and set aside.
In a small bowl, whisk together the rice vinegar, water, honey, and sea salt. Set this vinaigrette aside—it will be a delightful dressing for your bowl.
Heat the olive oil in a large skillet over medium heat. Add the thinly sliced carrots and sauté for about 2 minutes until they start to soften.
Add the sliced savoy cabbage and scallions to the skillet. Sauté for an additional 5-7 minutes until the cabbage is tender and slightly caramelized.
Pour the soy sauce over the vegetables and stir well to combine. Let it cook for another minute or two until everything is well coated.
In a separate pan, heat the ghee over medium heat. Crack the eggs into the pan and fry them to your desired doneness. For a runny yolk, aim for about 2-3 minutes.
To assemble the bowls, divide the cooked udon noodles among serving bowls. Top with the sautéed cabbage and carrot mixture, then place one or two fried eggs on top.
Drizzle the reserved vinaigrette over each bowl. Sprinkle with sesame seeds, scallion greens, and chili flakes if desired.
Serve immediately and enjoy the delightful combination of flavors and textures in your Cabbage Noodle Bowls with Fried Eggs!