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.
Start by creaming butter in a stand mixer with the paddle attachment - whip 5 minutes until fluffy and light. Slowly add powdered sugar. When it gets so thick its not frosting like add a splash of milk. Alternate this process until all the powdered sugar is added, and your frosting is the desired consistency.
Add 1-3 drops of red food coloring (depending on how pink you'd like the frosting) and whip on high for 3-5 minutes. The longer you whip it the fluffier off will be!
Frost chilled cookies. Eat immediately, or chill again before serving.
Keep cookies in a sealed container until they are ready to serve. Swig cookies taste great chilled. I even like to keep them in the freezer before transporting anywhere - makes it cleaner to transport and they'll taste great.