In a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, cream together the unsalted butter, powdered sugar, salt, and almond extract until smooth and well combined.
Gradually add the all-purpose flour, mixing on low speed until the mixture becomes crumbly.
Gently fold in the chopped maraschino cherries and continue mixing on low speed until a dough forms.
Transfer the dough onto a sheet of parchment paper. Use the parchment paper and your hands to shape the dough into a log approximately 2 inches in diameter. Wrap and chill the dough in the refrigerator for at least 1 hour.
Preheat your oven to 350°F (175°C). Prepare a baking sheet by lining it with a silicone baking mat or parchment paper.
Slice the chilled dough log into ½-inch thick pieces. Arrange the cookies on the prepared baking sheet, leaving space between each for slight spreading.
Bake in the preheated oven for about 12-14 minutes, or until the edges are set but not browned.
Remove the cookies from the oven and let them cool on the baking sheet for 2 minutes before transferring them to a wire rack to cool completely.