Place the mint leaves and simple syrup into a muddler positioned inside a sturdy glass or a mixing bowl. Gently muddle the mint until fragrant, pressing just enough to release the oils without tearing the leaves.
Fill a cocktail shaker with crushed ice from the 'ice' container, then pour in the muddled mint mixture along with the bourbon. Shake vigorously for about 15 seconds until well combined and chilled.
Fill a tall glass with crushed ice almost to the top. Strain the chilled mixture from the shaker into the prepared glass, ensuring the mint leaves are left behind in the shaker.
Garnish the drink with a sprig of fresh mint placed on top of the ice, providing a vibrant green contrast and aromatic finishing touch.
Serve immediately with a straw or stirring stick. Observe the cold, frosty appearance and the vibrant green garnish, signaling readiness to enjoy.