Cut the tomatoes in half crosswise, season with House Seasoning. Heat the oil in a small frying pan over medium heat. Add the garlic, and cook until just starting to turn golden brown, 1 to 2 minutes. Pour the garlic and oil into a heatproof bowl.
Oil the grill and set for high heat. Place the tomatoes cut side down and grill for 3 to 5 minutes. Turn over, top with the garlic oil mixture and continue to cook for another 3 minutes. Remove from grill and top with thyme, House Seasoning and Parmigiano-Reggiano. Serve immediately.
Mix ingredients together and store in an airtight container for up to 6 months.