Thinly slice the onion. Heat butter and oil over medium heat. Add onions and stir them gently.
Spread the onions out evenly over the pan and let cook, stirring occasionally. After about 10 minutes, they will start to smell caramelized. Check so they don't burn. Cook until the onions are a rich, browned color. Add salt, mix, and set aside.
Preheat the oven to 390F/200C. Prepare 2 baking sheets with parchment paper.
Unroll puff pastry and cut into 12 rectangles.
In a small bowl make an egg wash with 1 egg, and 1 tablespoon of water.
Add about a tablespoon of caramelized onion to the center, top with cheese, and a slice of pear.
Brush the corners with a little bit of water. Fold in one corner, then fold in over the second corner. Tightly pinch together so they don't unfold in the oven, then brush with egg wash. Repeat with the rest of the ingredients.
Bake for 20 minutes, or until golden brown. Let it rest for 5 minutes before removing them from the baking sheet.