Cut tomatoes into ¼" thick slices, sprinkle with salt and let it rest for 15-20 minutes to drain. Blot with paper towels and set it aside.
Turn oven to 400° and line baking sheet with parchment paper.
On a floured surface, roll out puff pastry into a 13" x 8" rectangle. Leave ¾" border around the edges.
Spread pesto sauce evenly over the puff pastry, then sprinkle shredded cheese and arrange burrata slices over the shredded cheese.
Add slices of baked eggplants over the cheese then arrange tomatoes over the eggplant slices. Brush the puff pastry border with egg wash and bake the tart for 18-20 minutes or until the puff pastry is golden brown.
Let the pastry cool for 10 minutes, then serve with basil leaves as garnish.