Heat oven to 350° F. Heat 1 tablespoon of the oil in an ovenproof nonstick skillet over medium heat. Add the scallions and cook, stirring, for 1 ½ minutes.
Add prosciutto and cook, stirring occasionally, until slightly crisp, 3 to 4 minutes.
Meanwhile, in a medium bowl, whisk together the eggs, milk, Parmesan, and pepper. Pour into the skillet and stir.
Bake until browned around the edges and puffed (a knife should come out clean), 15 to 20 minutes.
Meanwhile, divide the arugula among plates. Drizzle with the remaining oil, sprinkle with salt, and top with goat cheese.
Cut the frittata into triangles and serve with the salad.