Preheat oven to 350 degrees.
Cream butter, oil, sugars, water, extracts, and eggs in a stand mixer with the paddle attachment.
Separately, combine dry ingredients, then slowly add to wet. Mix until everything is combined.
Scoop or roll dough into 2-3 tbsp. sized balls. Lay out 9-12 dough balls, well spaced, on a cookie sheet. Chill for 15-20 minutes.
Now its time to give these cookies their signature rough edge. Put 1/4 c of sugar in a dish (this is in addition to the sugar listed above.) Stick the bottom of a glass in it. This is going to be your cookie press. Firmly press it into the center of your dough ball. You want your dough to spill out over the sides of the glass.
Bake at 350 for 7-8 minutes. They should just barely be set on the top (not even brown on the top). Remove from oven and let rest on pan. They will finish cooking and form on the pan.
Once they are cool put them in the fridge. They taste best if you frost them when chilled, but you can also eat/frost them at room temperature.