Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, baking soda, cinnamon, and salt in a medium bowl, stirring with a whisk.
Place 1/2 cup butter and 1 cup sugar in a large bowl; beat with a mixer at medium speed until well blended. Add egg and vanilla; beat well.
Add flour mixture; beat at low speed just until blended. Stir in apple and nuts.
Drop dough by tablespoonfuls 2 inches apart onto baking sheets lined with parchment paper. Bake at 375° for 10 minutes or until lightly browned. Cool on pans 5 minutes.
Combine 1/4 cup melted butter, 2 tablespoons sugar, and 1/2 teaspoon cinnamon in a small bowl. Lightly brush tops of cookies with the butter mixture. Cool on pans 10 minutes; remove from pans. Cool completely on wire racks.
Combine confectioners' sugar and milk in a small bowl; stir until smooth. Drizzle glaze over cookies.
Interesting Facts
These cookies are best served warm.
You can use any type of apple for this recipe, but tart apples, such as Granny Smiths, are the best option.
You can substitute walnuts for pecans or almonds for a different flavor.