Unfold a sheet of puff pastry on a lightly floured surface and roll it lightly to an 11 by 11-inch square. Using a 3-inch wide round cookie cutter, cut 4 circles from the sheet of puff pastry, discarding the scraps. Repeat with the second pastry sheet to make 8 circles in all. Place the pastry circles on 1 large sheet pan 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 place 1/8 of the onion mixture on each circle, again staying within the scored edge. Crumble 1/2 ounce of goat cheese on top of the onions. Sprinkle the diced tomato in the center of each tart along with the basil, salt, and pepper.

Bake for 20 to 25 minutes, until the pastry is golden brown. Serve hot or warm.