Defrost puff pastry according to package directions.
Peel, quarter, and thinly slice the onion. Peel and mince garlic.
Heat 1 T oil in medium sautee pan. Sautee onions and garlic over medium heat until the onions are soft and beginning to caramelize.
Season onions with salt and pepper, deglaze with wine, and stir in thyme.
Once the puff pastry is defrosted, unfold it onto a cutting board. Cut 2 6-inch rounds out of defrosted puff pastry. You can use a small plate or saucer and a paring knife for this. Put the puff pastry onto a baking sheet lined with parchment paper.
Score the puff pastry 1/2-inch from edge. This will allow the edge to rise more, creating a crust. The ingredients will go within this circle, leaving the edge empty.
Sprinkle the grated parmesan within the scored circle. Top with caramelized onions.
Crumble the goat cheese and sprinkle over the onions.
Slice the tomato in half. Cut 1 thick slice from each half of the tomato (1-inch thick). Place in the center of the tart, over the onions and cheese.
Drizzle 1-2 T olive oil over the tomatoes to encourage browning.
Julienne the basil, and sprinkle evenly over the tomato.
Put a few slices of parmesan on top of each tart and sprinkle with salt and pepper.