Using an electric mixer, beat the butter and sugar together until light and fluffy—5 minutes. Add the salt, pumpkin pie spice, eggs and vanilla bean seeds. Beat to combine.
Slowly add the flour until the dough in smooth. Then separate the dough and wrap it in 4 separate pieces. Place in the fridge and chill for at least one hour. (Could be made weeks in advance.)
When ready to bake, preheat the oven to 375 degrees F. Line several cookie sheets with parchment paper.
Remove one disk at a time. Flour a work surface and roll the disk out to 1/8- to 1/4-inch thick. Work fast and cut the cookies as close together as possible. The more times you roll out the dough the tougher it gets. Use a thin spatula to transfer the cookies onto the baking sheets. Repeat with remaining dough. Bake for 6-9 minutes. Cool completely on the cookie sheets before moving.
For the Sugar Cookie Icing:
Whisk the half & half, extract and powdered sugar together until smooth. Add food coloring if desired. Dip and face of each cookie into the icing, then set on a rack to dry.