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Preheat the Traeger to 375°F with the lid closed; this will take at least 15 minutes.
Place the puff pastry on a large rimmed baking sheet. Using a sharp knife, score the dough (without cutting through it) all around the perimeter about 1 inch from each edge; this creates a crust of sorts. Using a fork, prick all over the center of the puff pastry.
Place the sheet tray directly on the grill grate and parbake until the pastry has puffed and browned a little bit, 15 to 20 minutes.
While the pastry bakes, combine the cream, 3 of the eggs, both cheeses, and the chives in a small bowl. Season lightly with salt and pepper and whisk well to combine.
Remove the sheet tray from the grill but leave the grill on. Pour the egg mixture into the puff pastry. Lay the asparagus spears on top and sprinkle with the cooked pancetta, crumbling or chopping it if not already in pieces.
Whisk the remaining egg in a small bowl and brush the top of the edges of the pastry with it.
Return the tart to grill grate and cook until the egg mixture is just set, 15 to 20 minutes. The time will vary depending on the weather and your grill model.
Sprinkle the tart with lemon zest, more chopped chives, and shaved Parmesan to taste. Enjoy!