Preheat oven to 190C/375F/Gas 5 and line cookie sheets with parchment paper. Set aside.
Mince the cherries and squeeze out in a muslin or cheesecloth until very dry. Set aside.
Place the softened butter, oil, and sugars into a large mixing bowl and beat until combined.
Beat in the eggs, almond extract, cherry extract, and dry minced cherries.
Sift the flour, baking soda, and cream of tartar into a second large bowl. Slowly add one cup of the flour mixture at a time to the butter mixture beating immediately after adding it. Be careful to not over mix.
Using a small ice cream scoop to measure out even scoops of cookie dough, drop them onto the lined cookie sheets.
Bake for 8-11 minutes or until edges and bottoms of the cookies are lightly browned.
Remove cookie sheets from the oven and allow the cookies to cool for 5 minutes before transferring to a wire rack to cool completely.
To make the icing add the melted butter, cherry juice, almond extract, and cherry extract to a large bowl and whisk together.
Whisk in the powdered sugar one cup at a time until the icing is smooth, thick, and spreadable. If it runs off the cookies it is too thin.
Adjust as needed with additional cherry juice and/or powdered sugar to the mixture.
Spread the icing onto the cooled cookies and sprinkle with the slivered almonds.
Leave iced cookies to set before serving.
Store for up to a week in an airtight container. The cookies are at their best in the first few days, but they're so good, they probably won't last that long!