Preheat and prep: Heat oven to 350°F (175°C).
Line two baking sheets with parchment.
Whisk dry ingredients: In a bowl, whisk 2 1/4 cups flour, 1 teaspoon baking powder, 1/2 teaspoon baking soda, 1/2 teaspoon salt, 1 1/2 teaspoons cinnamon, 1/2 teaspoon ginger, and 1/4 teaspoon nutmeg.
Cream the butter and sugars: In a large bowl, beat 1/2 cup softened butter with 1/2 cup granulated sugar and 1/2 cup light brown sugar until fluffy, about 2 minutes.
Add wet ingredients: Beat in 1 large egg, 1 cup pumpkin puree, 1 teaspoon vanilla, and 2 tablespoons oil. Mix until smooth. The batter will look a little curdled—this is normal.
Combine: Add dry ingredients to wet and mix on low until just combined.
Do not overmix. The dough should be thick and spoonable, not runny.
Scoop: Drop rounded tablespoons or a 2-tablespoon cookie scoop onto the baking sheets, spacing about 2 inches apart. Smooth the tops with a damp fingertip for neater cookies.
Bake: Bake 10–13 minutes, until the edges are set and the tops spring back lightly to the touch.
They should not brown deeply.
Cool: Let cookies rest 5 minutes on the sheet, then transfer to a wire rack to cool completely before frosting.
Make frosting: Beat 6 ounces softened cream cheese and 3 tablespoons softened butter until smooth. Add 1 teaspoon vanilla and a pinch of salt. Gradually mix in 1 1/2 to 2 cups powdered sugar, beating until creamy.
Add 1–2 teaspoons milk or cream for a spreadable texture.
Frost: Spread a generous layer on each cooled cookie. Dust lightly with cinnamon or pumpkin pie spice if you like.