In a stand mixer fitted with the paddle attachment, beat the butter, brown sugar and granulated sugar on low speed to combine, about 1 minute.
Use a silicone spatula to scrape down the sides and bottom of the bowl, then add in the egg and vanilla. Beat until combined.
Stop the mixer and scrape down the bowl, then add the flour, graham cracker crumbs, baking powder, and salt and beat on low speed until just combined and no streaks of flour remain.
Use a 2 ounce cookie scoop to portion out equal amounts of dough, rolling them between your hands to smooth out the edges. Place on the prepared baking sheets, spacing the dough 2 inches apart. Press your thumb into the center of each dough ball pressing down to make a wide divot in the center. Place the cookies in the preheated oven and set a timer for 5 minutes.
While the cookies bake, make the pumpkin filling. In a small bowl combine the pumpkin, brown sugar, egg, evaporated milk, flour, and pumpkin pie spice. Whisk to combine.
Remove the cookies from the oven (they’re only partially baked!) and use the cookie scoop or a 1/4 cup measure to reshape the divot in the middle of the cookie. You should have a wide hole in the center and about 1/2 inch border of cookie on the outside. Spoon about 2 teaspoons of the pumpkin filling into the center of each cookie and place back in the oven for 9 to 11 more minutes or until the cookies are turning a light golden brown and the filling is no longer wobbly.
Remove the baking sheet from the oven and allow the cookies to rest on the baking sheet for 5 minutes, then use a metal spatula to transfer the cookies to a cooling rack to cool completely.
Once cool, top with a dollop of whipped cream if desired and serve!