Heat the oven to 350F (175C) and line a baking sheet with parchment.
Make the streusel. Stir the brown sugar, flour and pumpkin pie spice together with a fork. Work in the softened butter until the mixture holds together in crumbs, stopping when the largest clumps are about pea sized. Set aside.
Spread the pumpkin puree on a plate and press folded paper towels into it, swapping in dry ones until it has lost about half its volume and looks more like a soft dough than a puree. A fine mesh sieve works faster.
Put the blotted pumpkin in a large bowl with the cooled melted butter and both sugars. Mix until smooth. The mixture will go very dark, which is the brown sugar.
Add the egg yolks and vanilla and whisk until glossy.
Sift in the flour, pumpkin pie spice, baking soda, baking powder and salt. Fold with a spatula until you have a thick dough that holds its shape.
Scoop with a 2 inch scoop, spacing the balls about 2 inches apart, eight to a sheet.
Press a well into the middle of each ball and flatten slightly. Pack the streusel into the well and partway down the sides, using more than looks right, and press it in rather than sprinkling.
Bake for 12 minutes, until spread and lightly colored at the edges. The centers will still look soft. Leave on the sheet for 10 minutes to finish setting.
Whisk the powdered sugar, vanilla and milk together, slowly at first and then quickly to break up any lumps.
Once the cookies are cool, drizzle the glaze over parchment or a rack, since a fair amount runs off.