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. Your dough should be a little crumbly and not sticky at all.
Roll dough into 2-3 tbsp. sized balls. Lay out 9-12 dough balls, well spaced, on a cookie sheet.
Now its time to give these cookies their signature rough edge. Put 1/4 c of white sugar in a bowl (this is in addition to the 1 1/4 c. of white sugar in the dough). 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 browning on the bottom (not even brown on the top). Remove from oven right when you see the dough start to set (no brown!) and let rest on pan. They will finish cooking and form on the pan.
Move cookies to a cooling rack. Once they are cool put them in the fridge. Don't frost until cooled down to room temp.