Preheat oven to 400°F. On a lightly floured surface, roll out the puff pastry sheet into a 12-inch square. Cut into four 6-inch squares and place on a parchment-lined baking sheet. Prick each square several times with a fork.
In a large skillet over medium heat, melt the butter. Add the onions and cook, stirring occasionally, until golden brown and caramelized, about 20 minutes. Add the sugar and season with salt and pepper.
Divide the onion mixture among the tartlets. Bake until golden brown, about 20 minutes.