Preheat the oven to 350° Fahrenheit. Grease one 12-indent madeleine pan (or two 12-indent pans if you have them) and set aside. In a small saucepan, melt the butter over medium heat and stir constantly, continuing to cook, until the butter turns a light brown color and smells nutty or caramelly. Once it reaches light brown, take off the heat and pour into a heat-proof container to cool a bit (scrape any brown residue into the container as well). Let cool until just above room temp but not solid.
In a medium bowl, whisk together the dry ingredients for the madeleines: the gluten free 1-1 flour, arrowroot starch, baking powder, salt, and spices, then set aside. In a large bowl, using an electric mixer, beat together the eggs and the white and brown sugar, on high speed, until thick and lighter in color (about 3-4 minutes). Add in the pumpkin and vanilla and continue to beat until combined. Put away the mixer, then use a spatula to fold the dry ingredients into the pumpkin batter until smooth and combined. Lastly, slowly pour in the cooled browned butter and fold in using the spatula. Once combined, spoon one scant tablespoon of batter into each madeleine indentation. Do NOT fill past about 1/2 way or 2/3 of the way (leaving a solid 1/4 of an inch from the rim), or the batter will overflow and you won't have the iconic madeleine shape. Lightly tap the pan on the counter to release any tiny air bubbles in the batter.
Bake each tray for 10-12 minutes until firm and spongy to the touch but not too brown around the edges. Let cool completely before tipping them out of the tray onto a rack.
While the madeleines are baking, make the glaze. In a small bowl, combine all of the glaze ingredients at once and whisk together until smooth and combined, starting with 1 Tbsp of milk and adding more as needed for a thick, but still liquid/runny glaze. Set aside until madeleines are out of their trays and cooled, then dip each madeleine in the glaze or drizzle it over them. Then serve!