Preheat oven to 350°.
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. Remove from pans; cool completely on wire racks.