Preheat oven to 350 degrees.
Line baking sheets with parchment paper.
Beat butter and both sugars in the bowl of a stand mixer on speed 2 for 2 minutes.
Add egg and beat well.
Add pumpkin and 1 teaspoon vanilla; mix until smooth.
Add flour, baking soda, baking powder, 1 teaspoon cinnamon, nutmeg, and salt, mixing on speed 2 to combine.
Add walnuts and stir.
Drop rounded tablespoon of dough onto the prepared baking sheet, leaving a couple inches between cookies.
Bake for 12-14 minutes or until bottoms are golden and dough is fully baked.
Cool on baking sheet for 2 minutes, then transfer to wire racks to cool completely.
Combine powdered sugar, vanilla, cinnamon, 1 tablespoon butter, and 1/4 cup of milk in small bowl, stirring until smooth.
Add additional milk, 1 tablespoon at a time until the proper consistency is reached.
Spread glaze over cookies using a spoon or butter knife.
