In a shaker, gently muddle the mint leaves, simple syrup, and sugar cube until the sugar cube breaks apart. Add the remaining ingredients and whip, shaking with a few pieces of crushed ice, just until incorporated. Dump into a Collins glass and add crushed ice until the glass is about four-fifths full. Swizzle for a few seconds, then pack the glass with ice, mounding it above the rim. Garnish with the mint bouquet, placing it in the center of the ice and serve with a straw.