These are amazing - I've made them for years in different renditions and are a go-to easy but impressive dinner option. Equally good in summer or winter, these are a great way to do a fancy feeling dinner that is essentially pizza on store bought puff pastry crust. Just to clarify - cut TWO CIRCLES FROM A SHEET (the box of puff pastry comes with TWO sheets, making FOUR total tarts if you want to use both sheets and thus serve four people. You could make them smaller of course if you wanted but this is the way that Ina intended them sized, and they make quite a good dinner with a side salad. If they come out "heavy" you are doing something wrong.
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.