Heat olive oil and butter in a pan over medium heat. Add the sliced onions and cook, stirring occasionally, until they turn golden brown and caramelized. This should take about 20-25 minutes. Add sugar, salt, and pepper to taste. Stir in the fresh thyme leaves and set aside.

Roll out the thawed puff pastry on a lightly floured surface. Transfer it to a baking sheet lined with parchment paper. Using a sharp knife, score a border about 1 inch from the edge of the pastry. Prick the center of the pastry with a fork to prevent it from puffing up too much during baking.

Spread the caramelized onions evenly over the center of the puff pastry, staying within the scored border. Arrange the beefsteak tomato slices on top of the onions. Sprinkle the shredded mozzarella and grated Parmesan cheese over the tomatoes.

Brush the edges of the puff pastry with the beaten egg. Bake in a preheated oven at 375°F (190°C) for 25-30 minutes, or until the pastry is golden brown and the cheese is melted and bubbly.

Allow the tart to cool slightly before slicing. Garnish with additional fresh thyme leaves if desired. Serve warm and enjoy!