Preheat oven to 350 degrees F (175 degrees C). Oil a 9x5-inch loaf pan.

Place the onion and carrots into a saucepan, and pour in the beef broth. Bring to a boil over medium heat, and reduce heat to a simmer; cook, stirring often, until the vegetables are tender, about 8 minutes. Set the veggies and broth aside.

In a large bowl, beat the eggs, and stir in bread crumbs, basil, salt, and black pepper. Allow the mixture to stand for 5 minutes to moisten the crumbs, and lightly mix in the turkey, beef, and pork sausage. Mix in the cooked onions, carrots, and beef broth. Place the mixture into the prepared loaf pan. In a bowl, mix the tomato soup, balsamic vinegar, and Dijon mustard together, and spread the mixture over the meat loaf.

Bake until the loaf is no longer pink in the center and an instant-read meat thermometer reads at least 160 degrees F (70 degrees C), about 1 1/2 hours.