Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, nutmeg, and ginger. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and brown sugar until light and fluffy, about 2-3 minutes.
Add the egg and vanilla extract to the butter mixture, mixing until well combined.
Stir in the pumpkin puree until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
In a separate bowl, beat the cream cheese and powdered sugar until smooth and creamy.
Using a cookie scoop or tablespoon, drop a spoonful of cookie dough onto the prepared baking sheet, leaving space in between each cookie.
Make an indentation in the center of each cookie dough ball and fill it with about 1 teaspoon of the cream cheese mixture.
Cover the cream cheese filling with a little more cookie dough, sealing it in.
Bake in the preheated oven for 12-15 minutes, or until the edges are lightly golden. The centers may look slightly underbaked, which is fine.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.