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Place a piece of parchment paper underneath pot and use pencil to trace an outline around the edge. Cut out circle and cut a 2- to 3-in. hole in the center. Set aside.
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In pot over medium heat, whisk pear ingredients (except for pears and puff pastry). Peel pears, leaving the stem in place, and slice a thin portion off the bottom of each pear so they stand upright.
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Once mixture begins to simmer, turn heat to low and add pears. Cover pot with parchment paper round for roughly 18 minutes or until very soft.
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Transfer pears to cutting board, standing up straight, until completely cooled. Pour liquid into large bowl, add cooled pears and refrigerate overnight (optional).
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Remove pears from liquid. Pour liquid into pot over medium-high heat. Reduce liquid until thick and syrup-y. Pour syrup into container for serving.
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Preheat oven to 400 degrees F. and line baking sheet with parchment paper.
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Unroll thawed puff pastry sheet and slice into four squares, placing them on prepared baking sheet. Place pear on top of each square and cut around, removing any excess. Slice remaining sheet into ½-in.-wide strips. Beginning at the bottom, tightly wrap pear in about 2 strips, leaving stem exposed.
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In bowl, mix eggless “egg” wash ingredients together. Use brush to lightly coat puff pastry exterior. Bake for 25-28 minutes or until golden brown. Remove from oven and transfer to serving plate.
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Serve warm with a drizzle of reduced poaching liquid syrup or a dash of powdered sugar.