1. Preheat oven to 375 degrees F. Line a baking sheet with parchment paper.
2. In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger and nutmeg; set aside.
3. In the bowl of an electric mixer, cream together the butter and brown sugar on medium speed until light and fluffy, about 3 minutes. Beat in the eggs until combined. Reduce the mixer speed to low and add the pumpkin, half-and-half and vanilla. Beat until very well blended, about 2 minutes (the batter may look grainy at this point, but don't worry, it will come together). Keep the mixer on low and gradually add the dry ingredients, mixing until just combined. Use a rubber spatula to give it a final mix, incorporating any flour on the sides and scraping the bottom of the bowl.
4. Use a large cookie scoop to scoop balls of the batter onto the prepared baking sheets, leaving about 1½ inches between them. Bake for 12 to 15 minutes, or until the tops of the cookies spring back when lightly pressed. Transfer the baking sheet to a wire rack and let the cookies cool for 5 minutes then remove them to the rack to cool completely. Repeat with the remaining batter.
5. To make the icing: Add the powdered sugar to a heatproof bowl. Put the butter in a small saucepan over medium heet and let it start melting. Continue cooking, swirling occasionally, until the butter becomes golden brown and smells nutty (be paitent and watch carefully, it can go from brown to burnt quickly). Remove it from the heat and pour it over the powdered sugar, scraping the browned bits from the bottom of the pan. Add the half-and-half and vanilla and stir until the icing is smooth. Using a small offset spatula, spread icing on each cookie. (If the icing stiffens before you get them all frosted, stir vigorously to lighten it up.)
Note #1: You can substitute evaporated milk for the half-and-half in both the cookies and the icing.
Note #2: If you don't have a cookie scoop, transfer the batter to a pastry bag fitting with a plain tip (or use a ziploc bag with the corner snipped off) and pipe 1½-inch rounds onto the prepared baking sheet.