Preheat the oven to 400 degrees F (200 degrees C). Line 2 baking sheets with parchment paper.

Combine flour, sugar, baking powder, baking soda, and salt in the bowl of a food processor. Pulse 3 or 4 times to mix. Add butter and pulse until mixture resembles coarse crumbs, 20 to 25 times. Pour mixture into a large bowl. Fold in cherries and chocolate chips, making sure they are well coated with the dry ingredients.

Mix together 1/2 cup heavy cream, egg, and 1 tablespoon cherry syrup together with a fork in a small bowl until well combined. Stir into flour mixture and mix just until slightly moistened.

Turn dough out onto a well floured surface. Gather dough in a circle using floured hands, kneading only slightly to incorporate any remaining dry ingredients. Flatten dough into a 9-inch round disc; cut disc into 8 triangles. Transfer scones using a bench scraper onto the prepared baking sheets (4 scones per baking sheet). Brush scones with 1 tablespoon heavy cream.

Bake in the preheated oven until light brown, 20 to 23 minutes. Remove from the oven and let cool completely, about 30 minutes.

Mix powdered sugar and milk together in a small bowl. Transfer to a small resealable plastic bag. Snip one corner off of the bag with a pair of scissors, and drizzle glaze on cooled scones.