Line a large baking tray with baking parchment.
Roll out puff pastry on a floured surface to a 28x30cm rectangle and cut into six 10x14cm rectangles.
Refrigerate pastry rectangles on the tray for 20 minutes to firm up.
Preheat oven to 220°C (200°C fan) and mix cream cheese with lemon zest and seasoning.
Mark a 1cm border on each pastry piece, prick the centers with a fork, and brush borders with beaten egg.
Spread cream cheese inside the borders and top each with five 10cm asparagus spears.
Bake for 18-20 minutes until the pastry is puffed and golden brown.
Simmer eggs in 4cm of water for 3-4 minutes until whites are set but yolks remain soft.
Place a poached egg on each tart and garnish with parmesan, basil, salt, pepper, and olive oil.