Preheat oven to 350° F. Line 3 large cookie sheets with parchment paper.
In a medium bowl, whisk together the all-purpose flour, salt, baking soda and cream of tarter. Set aside.
In the bowl of a stand mixer, cream butter, 1 ¼ cups granulated sugar and the powdered sugar together until light and fluffy; about 4 minutes. Slowly add in oil while mixer is on. Add in eggs, one at a time, then add vanilla extract (or cake batter extract). Mix until just combined.
Gradually add flour mixture to the wet mixture until just combined.
Scoop two scoops of dough with a small cookie scoop. Roll both scoops into one ball then place on prepared baking sheets. (I fit 9 balls per cookie sheet)
Pour the remaining 1/4 cup granulated sugar into a small bowl. Take a drinking glass that is about 3 inches wide, and dip the bottom of the glass into water then dip into the sugar. Press the glass into each ball of dough to flatten into 1/2- inch thickness. (This will give the dough the jagged edges) Repeat process with all dough balls.
Bake cookies for only 8 to 9 minutes. Don't over-bake. (The short baking time is what makes these cookies soft!) Remove from oven and allow to cool on the baking sheets. No need to transfer to wire rack. Once cooled, store cookies covered in fridge until ready to frost and eat. (They can also be stored at room temperature.)