In a large bowl cream together butter, vegetable oil, 1 1/2 cups sugar, powdered sugar, water, and eggs.
In a medium bowl whisk together flour, baking soda, cream of tartar, and salt.
Add dry ingredients to wet ingredients and mix until combined. (dough should be crumbly and not sticky)
Roll dough into balls (slightly larger than a golf ball) and place on a lightly greased baking sheet about 3-4 inches apart.
Place remaining 1/4 cup sugar on a plate. Use the bottom end of a tall glass cup (or the bottom of a small jar) for pressing the cookies. Spritz the bottom of the glass/jar with cooking spray then dip the bottom into the sugar. Press cookies to about 1/2 inch thickness. Let the edges of the cookie dough squish out past the edges of the glass.
Bake at 350 for 8 minutes. Allow to cool for a few minutes on the baking sheet, then transfer to a clean flat surface or a cooking rack. When completely cool, place in airtight containers (if stacking, separate layers with parchment or wax paper) and keep chilled in the fridge.
For the frosting, cream together butter, sour cream (or greek yogurt), and cream cheese until smooth. Add remaining ingredients and mix until smooth. Store chilled in airtight container.
Immediately before serving, frost cookies. Top with shaved almonds if desired.
Store in airtight container in fridge up to five days.