Smash the mint in a shaker (or for a very minty drink, using a mortar and pestle), then add the Applejack. Ideally, let the mint steep for 30 minutes or so.
Add the lemon juice and simple syrup to the shaker. Fill the shaker with ice, and shake hard to further smash to the mint.
Pour over ice in a rocks glass (through a strainer if you have one). Garnish with a cherry.