Preheat the oven to 350°F (175°C).
Place the softened unsalted butter in a large mixing bowl. Using an electric mixer set to medium-high speed, beat the butter for approximately 30 seconds, or until creamy.
Add the granulated sugar, baking powder, baking soda, salt, and ground cinnamon. Continue beating until the ingredients are thoroughly incorporated, scraping down the sides of the bowl as necessary.
Add the eggs and vanilla extract. Beat again until evenly blended.
Mix in the pumpkin puree and continue beating until the batter appears smooth and uniform.
Gradually incorporate as much of the flour as the mixer can comfortably handle. If the dough becomes too thick for the mixer, use a wooden spoon to stir in the remaining flour until evenly combined. Avoid excessive mixing so the finished cookies remain tender.
Scoop the prepared pumpkin dough in generously rounded teaspoonfuls and arrange them on ungreased cookie sheets, leaving approximately 2 inches of space between each portion.
Bake in the preheated oven for 10 to 12 minutes, or until the tops appear set and lightly spring back when gently touched.
Transfer the baked cookies carefully to a wire cooling rack. Allow them to cool completely before adding the frosting so the topping does not melt.
Place the unsalted butter and packed light brown sugar in a small saucepan over medium heat.
Stir continuously as the butter and sugar melt together, continuing until the mixture is smooth. Remove the saucepan from the heat.
Transfer the warm brown sugar mixture to a medium mixing bowl.
Stir in the milk and vanilla extract until completely blended.
Gradually beat in the confectioners’ sugar until the frosting becomes smooth and easy to spread. Sifting the confectioners’ sugar beforehand can help produce a lump-free texture.
If necessary, incorporate a small splash of additional milk to loosen the frosting until it reaches a smooth, spreadable consistency.
Once the cookies have cooled completely, spread a generous coating of brown sugar frosting over the top of each cookie.
If desired, dust the frosted cookies lightly with ground cinnamon for additional warmth, flavor, and decoration.
Allow the frosting to set before serving.