Preheat oven to 425F. Line a baking sheet with parchment paper.
Place the puff pastry in the center of the baking sheet and brush all over with the egg. Form a crust by folding over the edges about a 1/2 inch, then brush the edges with egg. Use a paring knife to cut the folded edges and corners.
Sprinkle evenly with the Parmesan cheese, then poke the dough all over with a fork, making sure to go the whole way through the dough, otherwise it will puff up. Bake until golden brown and crisp, 15 to 20 minutes. Transfer to a wire rack and let cool.
Meanwhile, place the tomato slices on a triple layer of paper towels. Sprinkle with the salt and let sit for 30 minutes. When ready, sprinkle most of the shredded mozzarella evenly over the bottom of the crust. Press any excess moisture out of the tomatoes with paper towels, then layer the tomato slices evenly over the mozzarella. Whisk the olive oil and garlic together and drizzle evenly over the tomatoes. Sprinkle remaining little bit of the mozzarella on top. Bake until the crust is deep golden brown, 10 to 15 minutes.
Let cool on a wire rack for 5 minutes, then sprinkle with basil. Slide the tart onto a cutting board, slice into pieces and serve.