Set your oven to 350°F (175°C) and line a baking sheet with parchment paper or a silicone baking mat.
In a medium bowl, whisk together the dry ingredients: flour, baking powder, baking soda, salt, and pumpkin spice.
In a separate large bowl, cream together the softened butter and sugar until light and fluffy.
Add in the pumpkin puree, egg, and vanilla extract, mixing until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Be careful not to overmix.
Using a cookie scoop or spoon, drop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Gently flatten the dough balls slightly with the back of a spoon or your fingers.
Bake in the preheated oven for 10-12 minutes, or until the edges are lightly golden and the cookies are set.
Remove from the oven and allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.
In a small bowl, whisk together powdered sugar, milk (or cream), and a dash of pumpkin spice until smooth. Adjust the consistency as needed by adding more milk or powdered sugar.
If using a streusel topping, prepare it by mixing together flour, brown sugar, melted butter, and pumpkin spice until crumbly.
Once the cookies are completely cooled, drizzle or spread the glaze over each cookie.
Sprinkle the streusel topping over the glaze while it’s still wet, so it adheres to the cookies.
Allow the glaze to set for a few minutes before serving or storing the cookies.
Serve your pumpkin sugar cookies with a hot beverage and enjoy the flavors of fall!