To prepare cookies, combine 1/3 cup granulated sugar, brown sugar, and butter in a large bowl; beat with a mixer at medium speed until light and fluffy.
Add vanilla and egg; beat well. Lightly spoon flours into dry measuring cups; level with a knife.
Combine flours, baking powder, and next 4 ingredients (through 1/8 teaspoon nutmeg) in a medium bowl, stirring with a whisk.
Add flour mixture to sugar mixture; beat at low speed until well blended.
Shape the dough into 30 balls.
Place 2 inches apart on baking sheets coated with cooking spray; flatten cookies with bottom of a glass.
To prepare topping, combine 1 tablespoon granulated sugar, 1/4 teaspoon cinnamon, dash of nutmeg, and allspice in a small bowl.
Sprinkle evenly over cookies.
Bake at 350 for 9 minutes. Cool 10 minutes on pans.