1. Preheat the oven to 350°F. Line a baking sheet with parchment paper and set aside.
2. In a stand mixer fitted with the paddle attachment, beat the butter, granulated sugar, and 1/2 cup brown sugar on low speed to combine, about 1 minute.
3. Scrape down the bowl, add 1 egg and vanilla bean paste, and beat until combined.
4. Add 1 1/4 cups flour, graham cracker crumbs, baking powder, and salt. Mix on low speed until just combined and no streaks of flour remain.
5. Use a 2-ounce cookie scoop to portion the dough. Roll into balls and place on prepared baking sheet, spacing 2 inches apart.
6. Press your thumb into the center of each dough ball to make a wide divot.
7. Bake cookies for 5 minutes.
8. While cookies bake, whisk together pumpkin puree, 3 tablespoons brown sugar, 1 egg, evaporated milk, 1 tablespoon flour, and pumpkin pie spice in a small bowl.
9. Remove cookies from oven after 5 minutes and reshape the divots using the back of the cookie scoop or a 1/4 cup measure.
10. Spoon about 2 teaspoons of pumpkin filling into each cookie center.
11. Return cookies to oven and bake for 9 to 11 minutes, until cookies are golden and filling is set.
12. Let cookies rest on baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
13. Optionally, top cooled cookies with a dollop of whipped cream before serving.