Unfold a sheet of puff pastry on a lightly floured surface and roll it lightly to an 11 by 11-inch square. Using a 6-inch wide saucer or other 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 all. Place the pastry circles on 2 sheet pans lined with parchment paper and refrigerate until ready to use.
Preheat the oven to 425 degrees F.
Heat 3 tablespoons of olive oil in a large skillet over medium to low heat and add the onions and garlic. Saute for 15 to 20 minutes, stirring frequently, until the onions are limp and there is almost no moisture remaining in the skillet. Add 1/2 teaspoon salt, 1/4 teaspoon pepper, 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. Finally, scatter 4 or 5 shards of Parmesan on each tart.
Bake for 20 to 25 minutes, until the pastry is golden brown. The bottom sheet pan may need an extra few minutes in the oven. Serve hot or warm.
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