Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Place lemon halves, cut-side up, on a baking sheet.
Broil until lemon halves are blackened on the cut side, about 10 minutes. Allow lemons to cool enough to handle. Squeeze juice from the lemons into a measuring cup. Measure out 3/4 fluid ounce broiled lemon juice, reserving the rest for another use.
Pour sugar into a shallow bowl. Run lemon wedge around the rim of a cocktail glass; dip into sugar.
Combine whiskey, orange liqueur, and 3/4 fluid ounce broiled lemon juice in a cocktail shaker and add ice; cover and shake for 20 seconds. Strain drink into the prepared cocktail glass.