Put 3 tablespoons sugar in a 1- to 1 1/2-quart pan; shake pan often over high heat (do not stir) until sugar is liquefied and amber colored, about 3 minutes.
Remove from heat and add 3/4 cup dry red wine and 3 tablespoons balsamic vinegar (mixture will bubble vigorously and sugar will harden). Return to high heat and add 2 teaspoons minced garlic, 3/4 teaspoon black peppercorns, 3/4 teaspoon chopped fresh or dried rosemary leaves, and 1/2 teaspoon salt. Stir often until caramelized sugar melts again and mixture is reduced to about 3/4 cup, about 5 minutes.
Let stand for at least 2 hours.
Pour through a fine strainer set over a small bowl.
Whisk in 3 tablespoons extra-virgin olive oil and 2 teaspoons fresh lime juice.
