Fried Rice with Spring Vegetables and Fried Eggs

Name: Fried Rice with Spring Vegetables and Fried Eggs
Ingredients: 1 bunch ramps (about 5 ounces) or 2 bunches scallions, white and green parts separated; 2 tablespoons vegetable oil, plus more for drizzling; 1 small shallot, thinly sliced; ½ serrano chile, thinly sliced; 1 ½-inch piece ginger, peeled, grated; 1 large egg, beaten to blend; 3 cups leftover cooked rice or other grain; ½ bunch asparagus, sliced into 1-inch pieces on a diagonal; 1 cup shelled fresh peas (from about 1 pound pods) or frozen peas, thawed; 1 tablespoon (or more) white or regular soy sauce; 4 Olive Oil–Basted Fried Eggs (for serving)