Preheat oven to 375 degrees. Line 2 baking sheets with parchment paper.

Whisk flour, baking powder, baking soda, and salt in a medium bowl.

Beat eggs, pumpkin, brown sugar and oil in a bowl with an electric hand mixer until well combined, about 1 minute. Add 2 tablespoons of milk, 1 tablespoon pumpkin pie spice, 2 teaspoons espresso powder, and vanilla and beat until well combined, about 1 minute. Add the flour mixture in 3 parts, mixing until just combined after each addition.

Using about 1 1/2 tablespoons per cookie, drop rounded mounds of cookie dough onto the prepared baking sheets, at least 2 inches apart.

Bake cookies until firm and dry to the touch, 12 to 14 minutes. Let cool on the pan for 5 minutes, and then transfer to a wire rack to cool completely, about 20 minutes.

Beat the Neufchatel cheese, butter, and confectioners’ sugar in a bowl with an electric hand mixer until very smooth. Add the remaining 1 tablespoon milk and 1 teaspoon each of pumpkin pie spice and espresso powder to the Neufchatel cheese mixture until combined, about 30 seconds. Spread the frosting on the cookies, sprinkle with additional pumpkin pie spice or espresso powder, if desired.