Whisk together all filling ingredients — pumpkin puree, sweetened condensed milk, cinnamon, pumpkin spice, salt, and egg — until smooth. Transfer to a liquid measuring cup or piping bag without a tip for easy filling.

Whisk together the flour, baking powder, baking soda, kosher salt, and pumpkin spice in a medium bowl. Set aside.

In a stand mixer fitted with the paddle attachment, beat the butter on medium-low until smooth. Add both sugars and mix until a smooth paste forms. Scrape down the bowl and paddle, then mix again.

Add the eggs and vanilla extract. Mix on medium, scraping the bowl several times, until the mixture is lightened and evenly combined.

With the mixer on low, add the flour mixture one large scoop at a time, waiting until each addition is nearly incorporated before adding the next.

Scoop the dough using a ¼ cup cookie scoop onto parchment-lined baking sheets. Roll each portion into a ball, then press a well into the center of each using the back of a round tablespoon measure, a small cookie scoop, or a melon baller. Press deep but do not push through the bottom.

Refrigerate the shaped dough for at least 1 hour, or until fully chilled and firm.

Preheat the oven to 350°F conventional. Pour the pumpkin filling into the center of each chilled cookie, filling just to the top edge of the well.

Bake for 20 minutes, or until the edges are golden brown. Let cool on the pan before transferring.