Preheat oven to 400°F. Place thawed puff pastry on a cutting board and use a rolling pin to flatten it, smoothing out the seams.
Transfer puff pastry to a baking sheet lined with parchment paper. Use a knife to create a 1-inch border around the puff pastry, then use a fork to poke small holes (dock) across the center to discourage rising. Whisk an egg with 1 teaspoon of water and lightly brush over the border. Bake 20-25 minutes or until crispy and golden brown.
Thinly slice shallots, then add to a saucepan and drizzle with olive oil. Caramelize over low heat until soft and golden. Remove from heat and set aside.
Once cooled, spread stracciatella evenly over the center of the tart, then top with caramelized shallots and everything but the bagel seasoning. Slice tomatoes and arrange them over the top.
Finish it off with fresh basil leaves, salt, pepper, a drizzle of olive oil, and balsamic glaze. Optionally, you can garnish with a little extra everything but the bagel and a few fresh shallot slices for a pop of color.
Notes
For best results, allow the puff pastry to thaw in the fridge for 3-4 hours before rolling. If you're short on time, you can thaw at room temperature for about 30-40 minutes. If puff pastry becomes sticky from condensation, refrigerate it for 15-20 minutes.
Dock the center well. This discourages it from puffing excessively.
If the center rises too much, you can gently flatten it with the bottom of a mug or a spatula while it's still warm.
Let the pastry cool before adding the cheese to prevent it from becoming runny.
Store leftovers in an airtight container in the fridge for up to 2 days. Keep in mind that the pastry will gradually soften, so this tart is best when eaten fresh.