Thinly slice 3 medium onions.
Caramelize the onions with 4 sliced garlic cloves in 2 tablespoons of butter and a pinch of crystallized sugar. About 15 - 20 minutes.
Core tomatoes and slice in half. Place cut side down on rimmed baking sheet. Lightly salt and spray lightly with olive oil.
Roast tomatoes in 275 degree Fahrenheit oven for 2 1/2 hours. Cool completely. You can do this overnight.
Preheat oven to 425 degrees Fahrenheit.
Prepare caramel in 12 inch cast iron pan with 1/2 cup crystallized sugar, 4 tablespoons water and 1/4 teaspoon Sherry vinegar.
Arrange roasted tomatoes on top of caramel, face side down. Sprinkle 1 tablespoon of thyme leaves over tomatoes.
Arrange caramelized onions evenly over tomatoes. Sprinkle grated Parmesan cheese over onions.
Roll out the puff pastry sheet, cut a 13 inch round and drape over the cast iron skillet. Tuck edges of pastry into pan. Cut slits on top of pastry.
Bake for 25 - 30 minutes until crust is puffed and golden brown and delicious.
Remove from oven and let cool for 5 minutes.
Place large plate on top of skillet and carefully flip over. Return any tomatoes that are stuck to skillet back to tarte tatin.