Core pears from bottom, leaving stems intact. Peel pears; cut 1/4 in. from the bottom of each to level if necessary.
In a large saucepan, combine water, sugar, honey, lemon halves, cinnamon and cloves. Split vanilla bean and scrape seeds; add bean and seeds to sugar mixture. Bring to a boil. Reduce heat; place pears on their sides in saucepan and poach, uncovered, until almost tender, basting occasionally with poaching liquid, 16-20 minutes.
Remove pears with a slotted spoon; cool slightly. Strain and reserve 1-1/2 cups poaching liquid; set aside.
Preheat air fryer to 325°. Unfold puff pastry on a lightly floured surface. Cut into 1/2-in.-wide strips. Brush lightly with beaten egg. Starting at the bottom of a pear, wrap a pastry strip around pear, adding additional strips until pear is completely wrapped in pastry. Repeat with remaining pears and puff pastry.
Place pears in a single layer in greased air-fryer basket. Cook until golden brown, 12-15 minutes.
Meanwhile, bring reserved poaching liquid to a boil; cook until liquid is thick and syrupy, about 10 minutes. Place pears on dessert plates and drizzle with syrup. Serve warm.