Heat the oven to 275℉. Brush a 9-inch cake pan with oil and line the bottom with a circle of parchment paper.
Toss the tomatoes on a baking sheet with a little olive oil, salt and pepper. Arrange them skin-side down and place in the oven to dry for 45 minutes.
Meanwhile, cook the potatoes in boiling salted water for 25 minutes. Drain and let cool. Cut each potato into ½-inch-thick discs.
While the potatoes cook, sauté the onion. Heat a skillet over medium-high flame. Add 2 tablespoons of oil, then add the onions with a pinch of salt and lower the heat to medium. Stir as needed until soft and golden, 10 to 15 minutes. Remove to a bowl, reserving the pan.
Make a caramel: Cook the sugar and butter in the reserved onion pan over medium-high heat, stirring constantly with a wooden spoon. When melted and golden brown, pour the caramel into the lined cake pan, spreading it quickly to coat the base of the pan. The caramel will harden; don't worry, it will remelt in the oven.
Assemble the tart: Scatter the oregano leaves onto the caramel. Arrange the potato slices, cut-sides down, in the bottom of the pan. Nestle the tomatoes into the gaps between the potatoes, and top with the onions. Sprinkle generously with salt and pepper. Cover the vegetables with cheese slices.
Cut a puff pastry disc 1 inch larger in diameter than the pan and lay it over the tart filling. Gently tuck the edges down around the potatoes inside the pan. (At this stage you can chill the tart for up to 24 hours.)
Bake the tart at 400℉ for 25 minutes, then reduce the temperature to 350℉ and bake for 15 minutes more, or until the pastry is browned and cooked through.
Remove from the oven and let set for 2 minutes only. Hold an inverted plate firmly on top of the pan and carefully but briskly turn them over together, then lift off the pan. Serve hot or warm.