For the cookies: Drain the maraschino cherries, reserving the juice for the icing. Finely mince the cherries and pat/squeeze them dry. Set aside.
In a large mixing bowl, cream together the softened butter, oil, granulated sugar, and powdered sugar until well combined.
Beat in the eggs, almond extract, cherry extract, and minced cherries.
In a separate bowl, whisk together the flour, baking soda, and cream of tartar. Gradually add the flour mixture to the butter mixture, beating until just combined. Be careful not to overmix.
Line baking sheets with parchment paper or silicone mats. Drop the dough by teaspoonfuls onto the prepared sheets.
Bake at 375°F (190°C) for 8-11 minutes, or until the edges and bottoms are lightly browned.
Remove the cookies from the oven and transfer them to wire racks to cool completely.
For the icing:
In a medium bowl, whisk together the melted butter, maraschino cherry juice, almond extract, and cherry extract.
Gradually stir in 4 cups of powdered sugar, mixing until smooth. Adjust the consistency by adding more cherry juice or powdered sugar as needed until the icing is thick but spreadable.
Once the cookies are completely cooled, frost them with the cherry icing. Sprinkle with slivered almonds and add additional sprinkles if desired. Allow the icing to set before serving.