Heat the oven to 425F. Gently melt the butter in a saucepan or in the microwave and let it cool while you mix the dry ingredients.
Stir together the sugars, flour, spices, baking soda, and salt. Whisk the eggs into the cooled butter and add the vanilla. Stir the wet ingredients into the dry ingredients, stirring just until it comes together.
In a soup plate or shallow bowl, mix together the white sugar, cinnamon, and nutmeg. Form small 1 1/2-inch balls of dough and roll them in the cinnamon sugar. Place them on an unlined, ungreased baking sheet and flatten slightly.
Bake for 8 minutes then remove and let cool on the baking sheet for 5 minutes. Remove to a wire rack. The dough can be refrigerated for up to 5 days, well-wrapped. It can also be frozen in logs.