Brown the butter in a light-colored pan over medium heat, stirring occasionally, until golden brown bits form and it has a nutty aroma, about 6-8 minutes. (Don't walk away; it's important to keep a close eye on it so it does not burn. If it burns, you must start again.) Pour it into a large mixing bowl or stand mixer bowl, including the brown bits. Cool to nearly room temperature, about 15-20 minutes, give or take.
Place a folded paper towel on a plate or work surface. Place the pumpkin puree in the center. Blot the pumpkin puree with another paper towel to remove excess moisture. Repeat with additional dry paper towels until one comes away mostly dry (I use a total of 8 half-sheets). Set aside.
Preheat oven to 350°F. Line two baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking soda, baking powder, salt, and spices. Set aside.
Add the pumpkin puree, both sugars, egg and vanilla to the bowl with the browned butter. Beat until thoroughly combined.
Add the flour mixture and mix on low until just combined. Use a spatula to scrape down the sides of the bowl and incorporate by hand.
Scoop the dough using a 2-tablespoon scoop and place directly onto the prepared baking sheets, leaving about 2-inches in between; 6 to 8 cookies per sheet.
Bake 13-15 minutes until the edges are set. (Rotate the cookies 180 degrees halfway through if your oven has hot spots like mine.) Cool on the baking sheet for 5 minutes, then transfer to a wire rack.
While the cookies bake, make the glaze: brown 4 tablespoons butter until golden and nutty. Cool slightly. Whisk in powdered sugar, maple syrup, 2 tablespoons cream, and salt. Add more cream, 1 teaspoon at a time, to achieve your desired consistency. A thinner glaze is good for drizzling; a thicker glaze can be spread with a spatula; somewhere in between is good for dipping.