Heat olive oil in a pan over medium heat. Add the sliced onions, sugar, and a pinch of salt. Cook, stirring occasionally, until the onions are golden brown and caramelized, about 20-25 minutes. Stir in the balsamic vinegar and cook for another 2 minutes. Set aside to cool.

Roll out the puff pastry on a lightly floured surface to fit your tart pan. Transfer the pastry to the pan and trim the edges. Prick the base with a fork to prevent it from puffing up too much during baking.

Spread a layer of ricotta cheese evenly over the puff pastry. Top with the caramelized onions, spreading them out evenly. Arrange the tomato slices on top of the onions, overlapping slightly. Sprinkle with fresh thyme leaves, salt, and pepper.

Brush the edges of the 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 crisp.

Allow the tart to cool slightly before slicing. Serve warm or at room temperature, garnished with additional thyme if desired.