Combine all the ingredients except the Hamilton rum in a julep tin and fill the tin about halfway with crushed ice. Holding the tin by the rim, stir, churning the ice as you go, for about 10 seconds. Add more crushed ice to fill the tin about two-thirds full and stir until the tin is completely frosted. Add more ice to form a cone above the rim. Drizzle the Hamilton rum over the top of the drink. Garnish with the mint bouquet and serve with a straw.