Line a large baking sheet with a layer of paper towels. Lay tomato slices in a single layer on the baking sheet. Sprinkle both sides of tomato slices with salt. Allow the tomatoes to sit for 30 minutes.
Heat oil in a large skillet over medium heat. Add the onions and sauté them until they are evenly coated with oil. Cover the pan and reduce the heat to low. Let the onions steam for about 10 minutes, stirring occasionally. Season with salt and pepper. Continue to cook the onions until they are nicely browned and there is almost no moisture in the pan, about 10-20 minutes more.
Unfold one sheet of puff pastry and place it on a baking sheet lined with parchment paper. With the tip of a sharp knife, lightly score a border ¾ inch from edges (do not to cut all the way through the puff pastry). Spread the goat cheese on the pastry sheet up to the scored border line. Sprinkle grated Parmesan cheese over the goat cheese. Spread onion mixture on top of the cheese. Lay dried tomato slices on top of the onions, overlapping as necessary. Brush the tomatoes lightly with olive oil and sprinkle with salt and pepper. Scatter more Parmesan over tart. Bake for 20 to 25 minutes, until the pastry is golden brown. Sprinkle baked tart with julienned basil leaves and serve.