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.
Using a cookie scoop (2 tbsp works well but you can make them whatever size you'd like), scoop balls of cookie dough onto a pan.
Now its time to give these cookies their signature rough edge. Put 1/4 c of sugar and 1 tsp. of cinnamon in a dish (this is in addition to the sugar and cinnamon listed in the recipe.) Roll each ball of cookie dough in your hands to make a smooth ball, then roll into the cinnamon and sugar mixture. Place back onto cookie sheet.
Stick the bottom of a glass in the cinnamon and sugar mixture. 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.
Once all cookies are pressed out, bake at 350 for 6-7 minutes. It should seem like they're undercooked. You barely want them cooked so they stay soft. They will finish cooking on the pan when you pull them out. Place pan on stovetop or cooling rack as soon as they come out.
To make frosting, vigorously whip butter with powdered sugar, and enough heavy cream to bring the frosting to the right consistency. Start with a couple tablespoons and add more as needed. Then add the cinnamon, vanilla, and almond extract.
Once cookies are cooled, frost and enjoy! These cookies store well in the fridge or freezer.