In a medium bowl, whisk together 3 ½ cups all-purpose flour, 1 ½ teaspoons baking powder, 1 ¼ teaspoon ground cinnamon, ¼ teaspoon ground nutmeg, and ½ teaspoon salt.
Add ½ cup room temperature unsalted butter and ½ cup shortening, to a large bowl and use a hand mixer or a stand mixer fitted with the paddle attachment to beat them until combined. Add 1 cup granulated sugar and ½ cup packed light brown sugar and beat again until creamy.
Beat in 1 large egg, 2 teaspoons vanilla extract, and ¾ cup pumpkin puree. Gradually add the flour mixture to the pumpkin mixture and mix until just combined, scraping down the sides and bottom of the bowl as needed.
Cover the bowl tightly with plastic wrap and chill in the refrigerate for at least 3 hours, or overnight.
Once the dough has been chilled and you are ready to bake the cookies, preheat the oven to 350 degrees Fahrenheit and line 2 baking sheets with parchment paper.
Scoop the dough into 2-tablespoon portions and roll them into uniform balls. Place them on the prepared baking sheets, leaving 2 inches between them.
Bake for 11-13 minutes, until the cookies have puffed up and are light golden brown around the edges. As soon as they come out of the oven, use the back of a teaspoon measuring spoon to press down in the center of the cookie to make an indentation.
Let the cookies cook on the baking sheet for 5 minutes before transferring them to a cooling rack to finish cooling completely. While the cookies cool, prepare the cream cheese filling.
In a medium bowl, use a hand mixer to beat 4 ounces room temperature cream cheese and 3 tablespoons room temperature unsalted butter until smooth. Add 1 cup powdered sugar, 1 teaspoon vanilla extract, and 1 pinch of salt and mix until combined.
To prepare the chocolate drizzle, add ⅓ cup milk or semi-sweet chocolate chips to one small bowl and ⅓ cup white chocolate chips to a separate small bowl. Split 1 teaspoon shortening, in half, adding half to each bowl of chocolate.
Microwave each bowl of chocolate in 30-second increments, stirring between each increment until completely melted.
When the cookies are completely cool, fill the indentation of each cookie with cream cheese filling and drizzle the top with the melted chocolate. Allow the chocolate to set before serving.