Preheat oven to 350°F. Spread hazelnuts in a shallow pan. Bake until lightly browned, 8 to 10 minutes. Let nuts cool slightly, then rub off and discard most of the brown skins. Chop nuts finely and set aside. Reduce oven temperature to 300°F.
Line baking sheets with foil or wax paper. In medium bowl, combine egg whites, cream of tartar, nutmeg and cinnamon. Using electric mixer, beat at high speed until foamy.
Gradually add sugar, beating until mixture is stiff and glossy.
Using a rubber spatula, gently fold in vanilla extract, hazelnuts and orange peel. Drop by rounded teaspoons onto prepared baking sheets, spacing about 2 inches apart. Bake until cookies feel firm when touched lightly, 25 to 30 minutes. Remove to wire racks and let cool.
Store in an airtight container at room temperature up to 3 days.