Place olive oil in a large pan over medium heat. Add the onions and stir continuously until softened, about 8 to 10 minutes. Sprinkle in the salt and continue stirring another few minutes. Add the balsamic and half the thyme and reduce the heat to medium low. Cook the onions, stirring occasionally, until caramelized and golden, about 30 to 40 minutes. Remove from heat and set aside.
Preheat oven to 350°F. Light grease a large baking sheet or line with parchment paper.
On a lightly floured surface, roll out both squares of puff pastry, until about 1/2 cm thick. Place puff pastry on prepared baking sheet.
Top puff pastry with caramelized onions, leaving a 1/2 inch gap around the edges.
Bake in preheated oven for 25 to 35 minutes, or until the pastry is golden brown and flaky. Remove from the oven and sprinkle with remaining thyme. Let it cool for a few minutes, and then cut into small squares.