Preheat the oven to 350F and line a large baking sheet (or two) with parchment paper.
Add the softened butter and light brown sugar to the bowl of a stand mixer fitted with the paddle attachment. Beat it on medium speed until light and fluffy– about 2 minutes.
Next, add the pumpkin puree, egg, and vanilla. Mix again until well combined. Scrape down the sides of the bowl as needed. Add 1 1/4 cups graham cracker crumbs, along with the flour, 1 teaspoon of pumpkin pie spice, and baking soda. Mix until just blended.
Scoop out 2 tablespoon sized portions and roll them into smooth balls.
In a small bowl, stir together the remaining ½ cup graham cracker crumbs and ½ teaspoon pumpkin pie spice. Roll the cookie dough balls in the graham cracker crumbs and then place them on the parchment lined baking sheet (make sure to place them at least 2” apart). Press each dough ball into a ½” thick disc.
Bake them for 10-12 minutes, or just until the edges of the cookies are set.
Remove the cookies from the oven and allow them to cool completely on a wire rack.