Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper.
Heat the olive oil and butter in a large skillet over medium heat. Add the sliced onions and salt, then cook for 8 to 10 minutes, stirring occasionally, until softened and lightly golden.
Reduce the heat to medium-low and continue cooking the onions for another 15 to 20 minutes, stirring every few minutes, until deeply softened and caramelized. Stir in the balsamic vinegar and cook for another 2 minutes. Set aside to cool slightly.
Place the thawed puff pastry on a lightly floured surface and gently roll it into a rectangle if needed. Transfer it to the prepared baking sheet.
Using a sharp knife, lightly score a border about 1 inch from the edges without cutting completely through the pastry. Prick the center lightly with a fork to help prevent it from puffing excessively.
Whisk the egg with the water to make an egg wash. Brush the outer border of the pastry lightly with the egg wash, taking care not to let it run down the sides.
Spread the Greek yogurt evenly inside the scored border. Arrange the caramelized onions over the yogurt, then sprinkle evenly with Parmesan cheese and fresh thyme. Season with a little additional black pepper.
Brush the pastry border with another light layer of egg wash. Bake for 20 to 25 minutes, or until the puff pastry is deeply golden, crisp, and fully puffed around the edges.
Allow the tart to cool for about 10 minutes before slicing. Serve warm or at room temperature.