Heat an oven to 400°F. Line a baking sheet with parchment paper. Sprinkle both sides of tomatoes lightly with salt and place in a single layer on paper towels. Let them stand 15 to 30 minutes for paper towels to absorb some of their moisture.

Unfold the puff pastry sheet on a lightly floured work surface. Roll out into a 9-by-13-inch rectangle about ⅛ inch thick. Transfer the dough to the prepared baking sheet. Using a fork, prick (or dock) the pastry all over. Fold over ½ inch of the pastry all the way around the edge.

Arrange the tomatoes and feta cheese evenly on top of the pastry rectangle. Sprinkle evenly with the dill and some freshly ground black pepper. Drizzle with a little herb oil and parmesan.

Bake until the pastry is puffed and golden brown, about 20 minutes. Remove from the oven and transfer to a work surface. Cut the tart into pieces and serve.