Preheat the oven to 400°F. Line a baking sheet with parchment paper.
Bring a large pot of water to a boil. Line a baking sheet with paper towels. Add the asparagus and blanch for 30 seconds then immediately remove them from the water and arrange them on the paper towel-lined baking sheet to cool.
Lightly flour your work surface then unfold the puff pastry onto it and roll it into a 10x12-inch rectangle. Transfer the pastry to the lined baking sheet then using a sharp knife, score a rectangular border about 1 inch from the edges. Prick the center of the pastry with a fork then bake it for about 10 minutes or just until it is pale golden.
Remove the pastry from the oven and sprinkle it with the shredded gruyere and Parmesan cheeses then arrange the blanched asparagus spears on top of the cheese, trimming any ends as needed to fit across the pastry. Brush the entire pastry and asparagus with the olive oil then sprinkle it with salt and pepper.
Return the tart to the oven for about 20 minutes or until the asparagus are tender and the pastry is deep golden brown.
Remove the tart from the oven and let it cool. Drizzle it with the balsamic syrup then slice and serve.
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