Cut the two flatbread into 6 equal pieces. Lay the flatbread out on a board and brush the top of the flatbread with oil. Rub the top of the oiled flatbread with the garlic clove. Turn the flat bread over so it is oil side down on the board. Evenly spread the dijon mustard on the bottom half of the flatbread sandwich.
Lay the brie cheese slices evenly over the top of the flatbread with the mustard. Top evenly with fresh basil. Place a large tomato slice on top of the basil. Add 1 more layer of brie cheese on top of the tomato and then place the top portion of flat bread on the sandwich.
Heat the grill or grill pan over a medium heat until hot. Place the sandwiches on the grill and cook for 2 minutes, or until the sandwich has golden grill marks on the bottom and the brie is beginning to melt. Flip the sandwiches over and cook for another 2 minutes, or until golden. Remove from the grill and allow to sit for 5 minutes.