In a stand mixer bowl with paddle attachment, cream butter and sugar until light and fluffy.
Add egg, maraschino cherry juice, and almond extract; mix until combined.
In a separate bowl, whisk together flour, baking soda, and cream of tartar.
Gradually add the dry ingredients to the wet mixture on low speed until incorporated.
Gently fold in the quartered cherries and chopped almonds.
Place dough on parchment paper and form into two 8-inch rolls; wrap tightly in plastic wrap.
Refrigerate rolls for about 2 hours until firm.
Preheat oven to 375°F and line baking sheets with parchment paper.
Slice chilled dough into ¼-inch thick cookies and place on sheets about 2 inches apart.
Bake for 7-9 minutes until edges are lightly browned.