Preheat the oven to 350°F. Line 2 baking sheets with parchment paper.
In a large bowl, beat 1 cup butter and granulated sugar at medium speed with a mixer until creamy. Add the eggs, vanilla extract and rum extract, beating until combined. In a medium bowl, combine the flour, baking powder, 1 teaspoon pumpkin pie spice and salt. Gradually add the flour mixture to the butter mixture, beating until combined. Wrap the dough in heavy-duty plastic wrap and chill for 1 hour.
On a lightly floured surface, roll the dough to 1/4-inch thickness. Use a 4-inch pumpkin cookie cutter to cut out the dough. Place the dough on the prepared baking sheets. Bake until the edges are very lightly browned, 8 to 10 minutes. Let the cookies cool on the pans for 2 minutes. Then remove and cool completely on wire racks. Spread the Cinnamon Cream Cheese frosting on the cookies if using. Garnish with sprinkles.
Cinnamon Cream Cheese Frosting:
In a medium bowl, whisk together cream cheese, 1/2 cup butter and 1 tablespoon pumpkin pie spice with an electric mixer. Add in the powdered sugar in thirds. Makes about 1 1/2 cups frosting.