Place the pumpkin puree in a bowl lined with paper towels or a clean kitchen towel. Blot it gently to remove excess moisture, making sure not to squeeze it completely dry. You want to reduce the moisture so you'll end up with about 1 1/3 to 1 1/2 cups of pumpkin. Set aside until later. I like to do this step first, as it really helps the cookies hold their shape and not become too cakey.
Preheat your oven to 350°F (177°C). Line baking sheets with parchment paper or silicone baking mats and set aside. In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, pumpkin pie spice, ground cinnamon, and ground ginger until well combined.
Using a hand mixer or a stand mixer fitted with the paddle attachment, cream the softened unsalted butter with both brown sugar and granulated sugar on medium speed until the mixture is smooth and fluffy, about 2 minutes. Add the room temperature egg and mix on high until combined, about 1 minute, scraping down the bowl as needed. Next, add the maple syrup (or milk or orange juice), vanilla extract, and the blotted pumpkin puree from Step 1. Beat on high until fully incorporated; the mixture may look a little curdled—this is normal.
Add the dry ingredient mixture from Step 2 to the wet mixture from Step 3. Blend on low speed until a thick and sticky dough forms. Using a scoop or your hands, portion out about 1.5 tablespoons of dough per cookie, rolling if desired, and arrange them 3 inches apart on the prepared baking sheets.
Place the baking sheets in the preheated oven and bake for 14-15 minutes, or until the cookie edges appear set while the centers still look soft. Remove the cookies from the oven and let them cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. I honestly think these cookies taste even better on day 2, after their flavors have developed.
In a medium bowl, use a hand mixer or stand mixer fitted with paddle or whisk attachment to beat the softened cream cheese until smooth and creamy. Add the softened butter and beat until well combined. Mix in the powdered sugar, maple syrup, and ground cinnamon (about 1/8 teaspoon). Beat on low speed until the icing is smooth and creamy. Taste and add more cinnamon if you’d like. For a lovely depth of flavor, I like to use pure maple syrup in the icing instead of imitation syrup.
Once the cookies are completely cool, dip the tops into the prepared icing or use a knife to spread the icing onto each cookie. Allow the icing to set slightly before serving. Store leftover iced cookies tightly at room temperature for up to 2 days, or refrigerate for up to 1 week. Enjoy!