Roll out a sheet of puff pastry on a lightly floured surface to an approximately 28x28cm/11x11-inch square.
Cut out 2 circles using a 15cm/6inch wide saucer or other round object as a guide.
Repeat with the second pastry sheet to make a total of 4 circles.
Place the pastry circles on parchment-lined sheet pans and refrigerate.
Preheat the oven to 210°C/410°F/gas mark 7.
Heat a drizzle of olive oil in a large skillet over medium-low heat.
Add the sliced onions and garlic, and sauté for 15 to 20 minutes until the onions are tender and translucent.
Season with salt, pepper, and add the white wine and minced thyme.
Cook for another 10 minutes until the onions are lightly browned.
Remove from heat.
Mark a 1cm / 2-inch wide border with a knife around each pastry circle.
Prick the pastry inside the scored lines with a fork.
Sprinkle a tablespoon of grated Parmesan on each circle, staying within the border.
Spread 1/4 of the onion mixture inside each circle, keeping within the scored border.
Crumble 1/4 of the goat cheese on top of the onions.
Place a tomato slice in the center of each tart.
Brush the tomato lightly with olive oil and sprinkle with basil, salt, and pepper.
Toss 4 or 5 Parmesan shavings on each tart.
Bake for 20 to 25 minutes until the pastry is golden brown.