For the infused simple syrup: Heat water in a small sauce pan until boiling. Combine water with sugar in a pint-sized container. Add mint leaves, jalapeño, and ginger root to sugar and water mixture. Refrigerate for 48 hours.
For the Hot Julep cocktail: Muddle mint leaves and simple syrup in a Julep cup or short cocktail glass. Fill glass with crushed ice and add bourbon. Garnish with mint sprig and jalapeño slice, and serve.