In a large bowl, mix butter, oil, pumpkin, sugars, vanilla and eggs together until smooth.
Slowly add in all dry ingredients and mix together.
On a prepared baking sheet, spoon or scoop dough (about 1 1/2 Tbsp) about 2-inches apart. Next, take the bottom of a drinking glass and press down on the cookie dough/balls.
Bake cookies for 8-9 minutes.
Prepare the glaze:
Mix together the powdered sugar, water (depending on the consistency you want), the pumpkin pie spice and cardamom until smooth.
When the cookies are baked, allow to cool for 3-5 minutes and then transfer to a baking rack. Right away, spread the glaze (1 tsp or more) over each warm cookie.
Allow the glaze to set for 2 hours before serving.
Notes
TIP: To prevent the dough from sticking to the glass, press the bottom of the glass in white sugar before flattening cookies.