Bring honey, sliced ginger, and 5 cups water to a simmer in a medium saucepan over medium heat.
Reduce heat and cook at a bare simmer, stirring occasionally, until honey is dissolved, 20–25 minutes.
Strain through a fine-mesh sieve into a large measuring glass or a medium bowl (you should have about 4½ cups); discard solids.
Pour honey syrup back into saucepan.
Add blended Scotch, lemon juice, and single malt Scotch, if using.
Heat over medium-low until hot but not boiling.
Divide cocktail among mugs.
Run a toothpick or skewer through each piece of candied ginger and set over the mug’s rim to garnish.
