Preheat the oven to 400°F (200°C) and line a rimmed baking sheet with parchment paper.
Unroll the thawed puff pastry onto the prepared baking sheet. Using a sharp knife, lightly score a 1-inch border around the edges, then prick the center area multiple times with a fork to prevent it from puffing up.
Bake the pastry for 10 minutes until it begins to turn lightly golden. Remove from the oven and let it cool for 5 minutes.
Spread the Dijon mustard in an even layer over the center of the pastry, making sure to stay within the scored border. Sprinkle the shredded Gruyere cheese evenly over the mustard.
In a small bowl, whisk together the eggs, heavy cream, fresh thyme, kosher salt, and black pepper. Carefully pour this egg custard mixture over the cheese layer.
Arrange the trimmed asparagus spears in a single layer on top of the custard and cheese, alternating the tips and ends.
Return the tart to the oven and bake for 15 to 20 minutes, or until the custard is set, the cheese is bubbly, and the asparagus is tender.
Remove from the oven and let the tart rest for 5 minutes. Drizzle the top with balsamic glaze, slice into squares, and serve warm.