Preheat oven to 375*. Brush Sheet pan (9x13, 10x16, or half sheet) with olive oil. Roll out puff pastry on a floured surface to about 14x12. (does not have to be exact) Set on oiled sheet pan and build up edges a little. Pierce all over with fork. Bake for 15 mins, checking every 5 mins for puffiness and re-pierce in necessary. Remove from oven and immediately sprinkle with provolone cheese. Set aside. Can be made 1 hour ahead. Leave at room temp.
Increase oven temp to 425*. Peel tomatoes, (if they're from the garden, this should be easy. If not, drop in boiling water for 20 seconds, cool, then peel.) Slice into 1/4" rounds .Pat dry between double paper towels. Arrange tomatoes on top of crust. Season with salt and pepper. Top with parmesan and basil. Bake for 9 mins. Remove from oven and sprinkle with olive oil. Cut into squares, large for lunch, smaller for starter.