Heat the oven to 350°F (175°C). Cover a cookie sheet with parchment paper.
In a separate bowl, whisk together the 2 2/3 cups all-purpose flour, 1 tablespoon pumpkin pie spice, and 1/2 teaspoon salt until evenly distributed. Place the mixture aside.
Place the softened cream cheese, unsalted butter, granulated sugar, and packed brown sugar in a large mixing bowl. Beat at medium-high speed for approximately 3 minutes, or until the mixture becomes light and fluffy.
Scrape along the sides and bottom of the bowl, then mix again briefly so the ingredients are evenly blended.
Add the egg yolk and vanilla extract, beating until fully incorporated.
Add the reserved flour mixture in three separate additions, scraping the bowl between additions when necessary. Once combined, fold the dough several times with a spatula to eliminate any remaining pockets of dry flour. The finished dough should be thick.
In a medium bowl, combine the whole egg, brown sugar, table salt, pumpkin pie spice, whole milk, pumpkin purée, and vanilla extract.
Whisk thoroughly until the mixture is uniform and smooth. Set the pumpkin filling aside while shaping the cookies.
Portion the cookie dough into 2-tablespoon amounts, approximately 36 grams each, and roll every portion into a ball.
Coat each dough ball evenly in the coarse sugar, then place the balls on the parchment-lined cookie sheet with approximately 2 inches of space between them.
Use a 1-tablespoon measuring spoon to carefully press an indentation into the center of each dough ball.
Spoon 1 tablespoon of pumpkin filling into each prepared well.
Bake the filled cookies for 14–17 minutes, until they are set and their bottoms have developed a pale golden-brown color.
Remove the cookies from the oven and transfer them to a wire cooling rack. Allow them to cool completely.
If additional dough remains, repeat the shaping, coating, filling, and baking procedure. Allow the cookie sheet to cool before preparing each subsequent batch.
After the cookies have cooled completely, add a piped swirl or small dollop of whipped cream, if desired.
For an optional finishing touch, lightly dust the whipped cream with pumpkin pie spice or cinnamon before serving.