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Combine lamb, cheese, salt, pepper, 1 tablespoons rosemary and 1 tablespoon balsamic vinegar in a large mixing bowl. Form into 4 burgers.
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Heat a nonstick 12-inch skillet over medium high heat for 1 minute. Add oil, swirl to coat bottom of pan. Add burgers, cook until an instant read meat thermometer registers 160 F. (about 5 minutes each side). Remove patties to a platter; cover with foil to keep warm.
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Turn heat to medium low; add ⅓ cup balsamic vinegar, scrape up any browned bits from bottom of pan. Reduce sauce for 1 minute. Stir in remaining rosemary, beef broth and water mixture and honey. Simmer until sauce is thick and bubbly (about 5 minutes).
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Spoon pan sauce over burgers, or serve on dinner rolls, if desired.