In a medium bowl, beat the cream cheese with granulated sugar and 1 teaspoon of the vanilla until smooth. Place teaspoon-sized dollops of this mixture onto a parchment-lined baking sheet and freeze until solid, about 45 minutes.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, and pumpkin pie spice.
In a large bowl, cream together butter and brown sugar until fluffy. Beat in egg, pumpkin purée, and the rest of the vanilla extract.
Gradually mix the dry ingredients into the wet ingredients until just combined. Chill the dough for 1 hour.
Preheat the oven to 375°F. Take a tablespoon of cookie dough, flatten it, place a frozen cheesecake dollop in the center, and wrap the dough around it. Roll into a ball and place on a baking sheet.
Bake for 12-15 minutes, or until the edges are set. Allow to cool on the baking sheet for a few minutes before transferring to a cooling rack.