Unfold a sheet of puff pastry on a lightly floured working top and roll it lightly to an 11 by 11-inch square.
Using a 6-inch round object as a guide, cut 2 circles from the sheet of puff pastry, discarding the scraps.
Repeat with the second pastry sheet to make 4 circles in total.
Place the pastry circles on 2 baking sheets lined with parchment paper or coated with cooking spray and refrigerate until ready to use.
Preheat the oven to 425 degrees F.
Heat 2 tablespoons of olive oil in a large skillet over medium, once the pan is hot, turn the temperature to low heat, then add the onions and garlic.
Saute 18 to 20 minutes, stirring frequently, until the onions are limp and the skillet is almost dry.
Add 1/2 teaspoon salt, 1/4 teaspoon pepper or to taste, the wine, and thyme and continue to cook for another 10 minutes, until the onions are lightly browned.
Remove from the heat.
Using a sharp paring knife, score a 1/4 inch-wide border around each pastry circle.
Prick the pastry inside the score lines with the tines of a fork and sprinkle a tablespoon of grated Parmesan on each round, staying inside the scored border.
Place 1/4 of the onion mixture on each circle, again staying within the scored edge.
Crumble 1 ounce of goat cheese on top of the onions.
Place a slice of tomato in the center of each tart.
Brush the tomato lightly with olive oil and sprinkle with basil, salt, and pepper.
At the end, arrange 4 or 5 shards of Parmesan on each top of the tart.
Bake for 20 to 25 minutes, in the middle of the time, rotate the two baking sheets up to down and left to right, until the pastry is golden brown.