In a large bowl, use an electric mixer to beat the butter until creamy. Add the sugar and mix until incorporated. Mix in the flour, pumpkin pie spice and salt. Add the pumpkin puree and vanilla. Beat until everything is mixed well. (The dough may end being a bit sticky.)
Scoop the shortbread dough onto a floured surface. Sprinkle a bit more flour on top, split into two portions and gently roll into logs. Place the logs in plastic wrap or parchment paper, wrap and refrigerate for several hours until the dough is firm (can refrigerate overnight).
When ready to bake the cookies, preheat the oven to 350°F. Line baking sheets with parchment paper or silpat mats.
Using a non-serrated knife, slice the dough into ¼-inch-thick discs. If you're not getting the shape you want, gently shape the edges into a rounded shape with your fingers as you place the cookies on a baking sheet. Bake for 20 to 25 minutes, or until the cookies are lightly browned on the edges. Immediately move the cookies to a wire rack; and let cool completely.
ADD THE GLAZE:
In a medium bowl, whisk the powdered sugar and spice with enough milk to create a thick glaze. Drizzle the glaze over the cooled cookies, and let them cool.
Notes
Don't melt the butter for this recipe. It's really important that's it's just at room temperature and not too soft.
If you find the dough to be too sticky to work with while rolling, just sprinkle on a little more flour until it's not so sticky.
Topping alternative: combine the powdered sugar with the spice and just dust the cookies in spiced powdered sugar instead of glaze.