Place pumpkin puree in the middle of a kitchen towel. Gather up edges of kitchen towel and gently squeeze pumpkin over a sink or bowl until all the liquid is squeezed out. Set pumpkin aside.
Place butter and sugars in the bowl of a large stand mixer, fitted with a paddle attachment OR in a large mixing bowl with a wooden spoon. Mix on medium speed or by hand until creamy, about 2 minutes.
Beat pumpkin, vanilla, oats, flour, pumpkin pie spice, and salt into butter mixture until dough is just combined. Wrap dough in plastic wrap and chill 2 hours or up to 6 days.
Preheat oven to 350F and line a square 8-inch with parchment paper. Remove dough from refrigerator and pat in an even layer in pan. Prick dough all over with a fork.
Bake shortbread at 350F 25-30 minutes until a light golden brown. Allow shortbread to cool completely on a wire cooling rack.