Slice tomatoes to ½ inch thickness and place on folded paper towels for 30 minutes to absorb some of the water.
Thaw frozen puff pastry until you can unfold it. Use a rolling pin to to slightly flatten and seal seam. Place dough on parchment paper-lined baking sheet.
Preheat oven to 400 degrees. slice onion into ¼ slices and separate into rings.
Use a pizza cutter or knife to score around all four edges of the crust, being careful not to cut all the way through. You are creating a seam for the dough on the edges to rise above the center of the tart.
Brush surface of puff pastry with olive oil, making sure to give the edges a light coating.
Tear the Swiss cheese into quarters and place randomly, but evenly on the pastry. Sprinkle the grated Gruyere cheese evenly over the Swiss.
Arrange the tomatoes on top of the cheese , not overlapping. You will probably have extras.
Evenly sprinkle the tart with the salt and pepper.
Arrange the onion rings on top of the tomatoes. Add the basil leaves, hiding some of them under edge of the tomatoes.
Sprinkle the Parmesan cheese on top.
Bake the tart in a 400 degree convection oven if possible, but a regular oven will do. Allow tart to bake for 30 minutes. If the edges start to get too brown, cover them with strips of aluminum foil.
Remove the tart from the oven after 30 minutes. Use a scrunched up paper towel to absorb any standing liquid on the top of the tart. Water from the tomatoes will make the tart soggy if you don't remove it.
Put the tart back in the oven for an additional 12-15 minutes of bake time.
Remove tart from the oven and allow to cool until just warm. This is the best time to eat this tart. You can eat it hot or cold as well, but warm is best. Use a pizza cutter to cut into desired size. I like a big slab but smaller pieces make a great appetizer.
Wrap loosely in aluminum foil and bake at 350 degrees for a few minutes to re-heat.