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 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 puff pastry sheet 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. Arrange the beefsteak tomato slices on top of the onions, slightly overlapping them. Sprinkle the shredded mozzarella and 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 cheese is bubbly and lightly golden. Remove from the oven and let it cool for a few minutes before slicing.