Puree 3/4 cup oil, parsley, and garlic in blender. Add chives and cheese. Puree until smooth. Season with salt and pepper. (Pesto can be made 2 days ahead. Cover and refrigerate.)
Whisk eggs in large bowl to blend. Whisk in 3 tablespoons pesto. Place 2 slices prosciutto, overlapping slightly, across each of 4 plates. Heat remaining 1 tablespoon oil in large nonstick skillet over medium-low heat. Add egg mixture and cook until softly set, stirring constantly, about 4 minutes. Season to taste with salt and pepper. Spoon eggs atop prosciutto; fold prosciutto over sides of eggs.
