Preheat oven to 350℉. Line cupcake pan with cupcake liners.
Remove 12 cherries from jar for decoration if desired. Chop remaining cherries and set aside.
In a small sauce pan, cook down the cherry syrup to about one half it's starting volume.
In a large bowl of stand mixer, add flour, sugar, vegetable oil, baking powder, root beer concentrate and salt. Using the paddle attachment, mix on low until combined. Scrape down sides of bowl, then beat on medium low for 1 minute.
With the motor running add one egg at a time pausing in between until egg is thoroughly integrated. Scrape down sides of bowl, add chopped cherries then beat on medium low for 1 minute.
Pour half of the reduced cherry syrup into a measuring cup, add enough milk to equal ½ cup. With the motor running on the mixer, slowly add the milk into the batter. Scrape down sides of bowl, then beat on medium low for 1 minute.
Pour batter into the prepared cupcake pan. Bake until done, approximately 20 minutes.
While cupcakes bake, combine powdered sugar, butter, root beer concentrate and remaining reduced cherry syrup. Add milk a tablespoon at a time until it reaches desired consistency. Put in refrigerator until ready to use.
Remove cupcakes from oven and move to a cooling rack to cool completely.
When cool, ice cupcakes and decorate with reserved cherries, sprinkles, etc.