Step 1Combine the lime juice and sugar in a small heavy-bottom saucepan. Bring to a simmer over medium heat, then immediately reduce the heat to very low and simmer about 1 minute. Remove from the heat and set aside.
Step 2Divide the strawberries among 4 old-fashioned glasses. Sprinkle equal amounts of balsamic vinegar over them (it'll be just a few drops in each, about one-eighth teaspoon). Divide the Grand Marnier among them, and muddle the ingredients in each glass.
Step 3Fill each glass about two-thirds full with ice, then pour equal amounts vodka over the ice in each. Add one-half ounce of the simmered lime juice to each glass. Pour equal amounts of the wine into each glass, then stir gently and serve.