Preheat a grill to medium-high heat.
Cut the lemons in half and dip each lemon half in the sugar. Grill the lemon halves cut-side down until grill marks appear, about 6 minutes. Remove from the grill and let cool.
In a medium saucepan, combine the 1/2 cup sugar, 1/2 cup water, honey and rosemary. Place the saucepan on the grill and heat until the sugar and honey are fully dissolved, about 5 minutes.
Squeeze the cooled lemons into a pitcher or large punch bowl. Add the sugar and honey mixture to the lemon juice, along with the vodka, ice and water to taste.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.