Step 1: Preheat the oven

Preheat your oven to 200°C (400°F). Line a baking tray with parchment paper.

Step 2: Caramelize the onions

Heat butter and olive oil in a large frying pan over medium heat. Add the sliced onions and cook for 10-15 minutes, stirring frequently, until softened and golden. Stir in the sugar (if using) and balsamic vinegar, then cook for another 5 minutes. Season with salt and black pepper. Set aside to cool slightly.

Step 3: Prepare the puff pastry

Roll out the puff pastry on a lightly floured surface, if needed, and transfer it to the prepared baking tray. Use a knife to score a 1-inch border around the edges without cutting all the way through. Prick the center with a fork to prevent it from puffing up.

Step 4: Add the toppings

Spread the caramelized onions evenly over the center of the puff pastry, staying within the scored border. Sprinkle the cheese and fresh thyme leaves on top.

Step 5: Brush the edges

Brush the edges of the puff pastry with the beaten egg for a golden finish.

Step 6: Bake

Bake in the preheated oven for 20-25 minutes, or until the pastry is golden and puffed, and the cheese is melted.

Step 7: Serve

Allow the tart to cool slightly before slicing. Serve warm or at room temperature.