Slice tomatoes into ¼ inch slices and salt both sides. Place on a cooling rack for 30 minutes (up to an hour) to remove excess moisture from tomatoes.
Preheat grill to 400 degrees Fahrenheit, and place pizza stone per your grills directions as the grill warms up.
Using a extra fine grader, grade garlic clove and mix with olive oil in a small dish.
Roll out puff pastry to a 12" by 16" rectangle and brush with the garlic oil mix, leaving 1 ½ inches along the edges unbrushed as that will be folded over onto the tomatoes.
Sprinkle mozzarella cheese into the center of the galette and then add tomatoes in layers. Top with Parmesan cheese and basil (again, leaving 1 ½ inches along the edge with no ingredients).
Place onto the pizza stone in the grill for 10 - 12 minutes or until the puff pastry is golden brown and tomatoes are cooked. Remove with a pizza peel and serve warm. Top with a pinch of finishing salt.