Start by browning the butter. You'll take ½ cup of butter (one stick) and melt in a medium saucepan, and then continue to cook past the point of boiling. You will do this over medium heat, past the point of the butter foaming, until it turns a nice golden brown, and has a nutty smell. Make sure you are stirring and watching it, because it will turn from perfectly brown to burnt fairly quickly. Once finished browning, pour all of it into a bowl and allow to cool. Once cool stir in 1 tablespoon of espresso powder.
Preheat oven to 350 F. Line two baking sheets with parchment paper.
Measure out ⅓ cup of pumpkin and place on a paper plate. Using some paper towels, blot the pumpkin, removing liquid until the paper towel wipes clean. When done the pumpkin should measure more like ¼ cup (60g).
Place browned butter and sugars in the bowl of a stand mixer, fitted with a paddle attachment and mix together. Add pumpkin, egg yolk and vanilla and mix again.
Add salt, baking soda, cinnamon, pumpkin spice and flour to the bowl and mix on low, being careful not to over mix. I usually like to stop mixing when I still have flour remnants around the bowl. Fold in toffee bits.
Scoop the dough into large balls, I like to use a 3 tablespoon cookie scoop. Press down slightly on the cookie ball so it's not quite as tall (or leave as is if you prefer a thicker cookie). Bake the cookies for 9-11 minutes. You don't want to see any shiny dough in the middle of the cookie, but make sure not to over bake these.
Mix together all the ingredients for the espresso glaze. You can add a bit more water or powdered sugar if needed to get desired consistency. Brush or drizzle the glaze over the top of the cooled cookies.