Combine tomato, Parmesan cheese, basil, garlic, vinegar and 2 teaspoons olive oil in a medium bowl. Set aside.
Season burgers on both sides with seasoning. Grill, turning once, until desired doneness or until the internal temperature reaches 160°F about 12 to 15 minutes.
Lightly brush both sides of the sourdough bread with olive oil. Grill 3 to 4 minutes, turning once, until bread is lightly toasted. Place burgers on bread; top with tomato mixture and additional slice of bread.
Recipe alternative: Eliminate one slice of bread on each burger and serve open-faced.