Make the honey-ginger syrup: Combine honey, ginger, and 1 cup water in a 2-qt. saucepan over high; boil. Reduce heat to medium and simmer 5 minutes; chill overnight, then strain, discarding solids.
Make the cocktail: Combine blended scotch, lemon juice, and syrup in a cocktail shaker filled with ice; shake vigorously and strain into a rocks glass with one large ice cube. Top with Laphroaig and garnish with candied ginger, if you like.