Heat the olive oil and butter in a large skillet over medium heat. Add the sliced onions and cook, stirring occasionally, until they begin to soften. Sprinkle with sugar and continue cooking, stirring frequently, until the onions are golden brown and caramelized. This process should take about 20-25 minutes. Stir in the thyme leaves and season with salt and pepper. Set aside to cool.

Preheat your oven to 375°F (190°C). Roll out the thawed puff pastry on a lightly floured surface to fit your tart pan. Transfer the pastry to the pan and trim any excess edges. Prick the base of the pastry with a fork to prevent it from puffing up too much during baking.

Spread the caramelized onions evenly over the puff pastry base. Arrange the sliced beefsteak tomatoes on top of the onions, slightly overlapping them. Sprinkle the grated Parmesan cheese over the tomatoes. Brush the edges of the pastry with the beaten egg to create a golden crust.

Place the tart in the preheated oven and bake for 25-30 minutes, or until the pastry is golden brown and the tomatoes are tender. If the edges of the pastry begin to brown too quickly, cover them with aluminum foil.

Allow the tart to cool for a few minutes before slicing. Garnish with additional fresh thyme leaves if desired. Serve warm or at room temperature.