Preheat the oven to 350 degrees F. Line a large baking sheet with parchment paper or a silicone mat.
In a large bowl, whisk together the flour, baking soda, cream of tartar and salt.
In a large bowl, use an electric mixer to combine the butter, oil, 1/2 cup of the granulated white sugar, and the powdered sugar until smooth. Add the egg, vanilla and almond extract. Mix until combined. Add the dry ingredients and mix on low until just combined.
Roll the dough into about 24 balls (about 2 tablespoons of dough per cookie). Arrange on the prepared baking sheet about 2-inches apart. Not all of the cookies will fit on the sheet for one batch.
Pour the remaining 1/4 cup granulated white sugar on a small plate. Spray the bottom of a drinking glass with nonstick cooking spray. Press the bottom of the glass into the sugar so that it's evenly coated with sugar. Press the sugared glass bottom on top of each cookie dough ball and press down gently, dipping the glass back into the sugar after each cookie so the glass doesn't stick to the dough and every cookie has an evenly sugared top.
Bake the cookies for 8 minutes. Do not overbake! The cookies won't look done- they will still be light in color and soft in the center, and that is okay. They will set up as they cool on the baking sheet. Let them cool for 5 minutes before transferring to a wire cooling rack to cool completely. When the baking sheet has cooled completely, repeat with the remaining dough.
MAKE THE FROSTING:
In a large bowl, use a mixer to beat the butter until smooth. Add the powdered sugar and mix until combined. Stream in the cream, vanilla and almond extract. Beat until the frosting is smooth. If the frosting is too thick, add a little more cream. If it's too thin, add a little more sugar.
ASSEMBLE THE COOKIES:
Use a butter knife to spread the frosting over the tops of the cooled cookies. Decorate with sprinkles, if desired. Store in an airtight container at room temperature for up to 3 days.