Peel the pears with a vegetable peeler, being careful to leave the stems intact. Slice off the bottom of the pears to make them stand up straight.
Using a large pot, combine the honey, sugar, water, and cinnamon sticks. Add the pears. Bring to a simmer and allow the pears to cook on medium-low until they start to soften.
While the pears are poaching, place the sheet of puff pastry on a piece of parchment or wax paper sprinkled with a small amount of flour. Roll the pastry gently to smooth out the wrinkles. Cut the pastry into long strips 1/2" wide.
In a small bowl combine the cinnamon and sugar. Sprinkle about 1-2 teaspoons of the cinnamon-sugar on the pastry strips.
Preheat the oven to 400℉.
Wrap the pears, starting at the bottom, with the strips of the puff pastry al the way to the top. Press the pieces together at the seams. Combine the egg and water in a small bowl. Brush each pear with the egg wash and sprinkle with more cinnamon-sugar (about 1 teaspoon).
Place the pears on a parchment-lined baking sheet. Bake for 15-20 minutes or until the pastry starts to turn golden brown.
Remove from the oven and serve with a scoop of cinnamon ice cream. Drizzle with salted caramel sauce.