Bring a medium pot of well-salted water to a boil. Meanwhile, prepare an ice water bath. Blanch asparagus until just tender, about 1 minute. Drain and transfer to ice water bath to cool. Drain. Set spears aside.
Whisk eggs with 1 tsp each salt and pepper. Stir in scallions, chives, and asparagus stalks.
Preheat oven to 425 degrees. Heat oil in an ovenproof 10-inch skillet over medium heat. Pour egg mixture into skillet. Cook until edges begin to set; push cooked edges toward the center of the pan with a wooden spoon. Continue cooking until the center begins to set, about 2 minutes more. Dollop goat cheese over top and sprinkle with reserved asparagus spears.
Bake until eggs are completely set, 8 to 10 minutes. Let cool slightly before serving.