Line baking sheet or plate with parchment paper. In a small bowl, use a whisk or mixer to mix the cream cheese with powdered sugar until smooth and combined. Place 16 heaping teaspoons of cream cheese on the baking sheet or plate. Slightly flatten each ball of cream cheese. Freeze for at least one hour, or until frozen almost completely solid.
When the cream cheese is frozen almost solid, line two large baking sheets with parchment paper and set aside. Add melted butter to a large mixing bowl, then whisk in the brown sugar and granulated sugar. Add in the pumpkin puree and vanilla extract. Mix until smooth.
Add in the flour, salt, baking soda, and pumpkin pie spice and stir until just incorporated. The dough will be a bit wetter than a typical cookie dough - do not add more flour!
Gently stir the chocolate chips and and graham crackers into the dough.
Preheat the oven to 375°F. Use a large cookie scoop (about 2½-3 tablespoons of dough) and scoop balls of cookie dough onto each baking sheet, allowing lots of room between each ball of dough.
Remove the cream cheese balls from the freezer and gently press one into the centre of each cookie. Cover the cream cheese ball with the dough, wrapping it around the cream cheese to cover it completely (photos are in post above for reference). Sprinkle more chocolate chips and/or graham cracker pieces on top if desired.
Once you've finished assembling one baking sheet of cookies, place them in the fridge while the oven finishes preheating, then repeat the steps with the second batch of cookies.
Once the oven has preheated, bake the cookies one baking sheet at a time for 9-12 minutes, or until the cookies are slightly brown on the edges and the top looks set. 11 minutes was perfect in my oven. If you leave the cookies in the oven for too long, they will become very brown, and the cream cheese will melt into the cookie dough! If you're unsure, you can bake one test cookie to start.
Let the cookies cool on the baking sheet for 10-15 minutes (they will continue baking a little bit during this time). Enjoy!