In a large skillet over medium-high heat, using the olive oil, crumble and brown the turkey burger, grated carrots, and onion, until turkey is no longer pink inside. Spread the turkey out to make room in the center of the pan and cook the minced garlic for about 1 minute.
stir the chicken broth and tomato sauce into the turkey mixture, followed by the hot sauce, red wine vinegar, Worcestershire sauce, brown sugar, and then salt and pepper to taste.
Cook until heated through and bubbly. Serve over buns with blue cheese sprinkled on top.