Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a large bowl, cream together the softened butter and sugar until light and fluffy.
Mix in the pumpkin puree, egg, and vanilla extract until well combined.
In another bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
In a separate bowl, prepare the cheesecake filling by combining the softened cream cheese, sugar, and vanilla extract. Beat until smooth and creamy.
Scoop out a tablespoon of cookie dough and flatten it slightly in your hands. Add a teaspoon of the cheesecake filling to the center, and then top with another tablespoon of cookie dough. Seal the edges and roll gently into a ball.
If desired, roll each cookie ball in the cinnamon-sugar mix for added sweetness and a slight crunch.
Place each cookie onto the prepared baking sheet, leaving space between each one.
Bake for 12-15 minutes or until the edges are set and the tops are slightly firm.
Allow the cookies to cool on the baking sheet for about 5 minutes before transferring to a wire rack to cool completely.