To make the bars:
Preheat the oven to 350°F. Line a 8 by 8-inch baking pan with foil or parchment paper, leaving an overhang.
In a large bowl combine the flour, cinnamon, ginger, nutmeg, baking soda and salt.
In another medium bowl whisk together the butter, brown sugar, egg, vanilla, and pumpkin. Add the butter mixture to the flour mixture, stirring until just combined. Pour the batter into the prepared pan.
Bake for 25 minutes, or until a cake tester inserted into the center comes out clean. Be careful not to overbake. Set the pan on a wire rack to cool completely.
To make the frosting:
In a large bowl, use an electric mixer to beat the butter and brown sugar on medium-high speed until light, fluffy, and smooth. Add in the cinnamon and salt. On low speed gradually add in the powered sugar until incorporated. Add in the vanilla and milk and increase the speed to medium-high. Beat until very light and fluffy. If too runny, add more powered sugar, If too thick, add more milk. Spread evenly all over the pumpkin bars.
Cut into squares before serving. Store in an airtight container at room temperature for up to 1 day or in the fridge for up to 5 days.