Whisk together the flour, sugar, salt, and yeast in the bowl of a mixer.
Add the butter, egg, and milk and mix with the hook attachment.
When the dough is smooth, remove bowl from mixer, cover and leave in a warm place for 20 minutes.
Return the bowl to the mixer and knead again using the hook attachment until the dough is smooth and elastic.
In a separate bowl, whisk flour, sugars, and pumpkin pie spice together.
Add butter and rub the ingredients together with your hands until it becomes the consistency of streusel crumbs.
In a saucepan heat sugar and water together whisking until sugar is dissolved and the mixture is smooth.
Keep mixture warm until ready to use so it doesn't harden and crack.
Preheat oven to 350 degrees F. Line two baking sheets with parchment paper.
Sprinkle flour on a work surface. Divide the dough into 10 equal pieces. Use a kitchen scale to do this, if you have one.
Take a piece and roll into a ball between your hands. Use a rolling pin to roll out a circle 5" in diameter. The thickness should be less than a raw sugar cookie cutout but not paper thin. Somewhere in between.
Place on a baking sheet and repeat with the rest of the dough.
Remove about 3 tablespoons worth of streusel from the bowl. From the main portion, pile a generous amount of streusel on each disk, spreading it evenly around.
Stir the pumpkin pie spice and salt into the pumpkin puree. Spoon the pumpkin mixture onto the streusel, spreading it out slightly. Sprinkle the reserved streusel crumbs on top.
Place in the preheated oven and bake for about 13 minutes or until the edges just start to turn golden.
Transfer to a wire rack and set the rack over the parchment lined baking sheet. Let cool completely.
Drizzle the glaze over the tops. Let the glaze harden, then plate and serve. These are best if eaten the day they are made.
*For the roasted pumpkin purée: