Heat a wok over medium-high heat until smoking. Pour in 1 tablespoon of the oil. Add the mushrooms and eggs and stir continually until the eggs are softly cooked, about 20 seconds, and then transfer the mushrooms and eggs to a bowl. Add the remaining 2 tablespoons oil to the wok and heat until hot. Add the rice, season the mixture with the five-spice powder and stir-fry until heated through.
Return the mushrooms and eggs to the wok with the Pickled Vegetables and stir continually until very hot. Season the mixture with the sesame oil and soy and sprinkle with salt and white pepper. Toss well, and then transfer the mixture to a serving platter. Garnish with the ramps and sprinkle some chili powder over the top. Serve warm or at room temperature.
Whisk together the mirin, rice vinegar, sugar and salt together in a small bowl until dissolved.
To pickle cucumbers: Slice the cucumber into half-moon slices and transfer to a nonreactive bowl. Pour the pickling liquid over them, toss to coat, and let stand for at least 30 minutes or up to 3 days in the refrigerator.
To pickle baby carrots: Peel and halve them lengthwise and transfer them to a nonreactive bowl or jar. Pour the pickling liquid over them, making sure they are completely immersed. Let stand for at least 1 day and up to 5 days in the refrigerator before serving.
To pickle radishes: If using breakfast radishes, halve them lengthwise, or if using round, quarter them lengthwise. Transfer them to a nonreactive bowl or jar and pour the liquid over them. Let stand for at least 2 hours or up to 2 days in the refrigerator before serving.