Preheat oven to 350 degrees F. Spray two large baking sheets with olive oil cooking spray.
Begin by making the cookie dough. Whisk together gluten free flour, salt, baking soda, baking powder cinnamon and nutmeg in a medium bowl until fully combined. In a mixing bowl, combine butter and sugars. Beat until light and fluffy. Add the egg, pumpkin and vanilla and mix until evenly combined. Add the dry ingredients in 3 batches and mix until just combined. Cover dough with plastic wrap and refrigerate for 30 minutes.
Next, make the cheesecake filling. Combine softened cream cheese, sugar and vanilla in a small bowl. Mix well until the mixture is smooth. Cover with plastic wrap and refrigerate for 30 minutes.
To make the cookies, begin by scooping about a small ball of the cookie dough and flatten into a pancake-like circle. Next, spoon a teaspoon of the cheesecake filling in the center of a cookie. Take a second cookie disk and place on top of filling. Pinch the edges together to seal, then roll the filled dough into a ball.
Roll in cinnamon-sugar and place on baking sheet. Repeat until all dough is use, spacing the cookies about 2” apart. Flatten the cookie dough balls slightly with your hands.
Bake for 10-15 minutes, until the tops start to crack. Rest on baking sheets for at least 5 minutes before transferring to a wire rack to cool completely.