In a 10-inch/25 cm oven-safe skillet, melt the demerara sugar and butter over medium heat. Let the mixture bubble gently for 3 to 4 minutes, until the sugar has fully dissolved and turns golden.
Add the thyme, salt, and white wine. Simmer for 3 to 5 minutes, allowing the liquid to reduce slightly and thicken into a light syrup.
Add the cherry tomatoes and stir gently to coat them in the glaze. Cook over low heat for 10 minutes, until the tomatoes begin to blister and release some of their juices. Remove the skillet from the heat and discard the thyme sprigs.
On a lightly floured surface, roll out the puff pastry into a round just larger than the skillet. Trim as needed and place it over the tomatoes, tucking the edges down between the filling and the sides of the pan.
Transfer the skillet to a preheated 355°F/180°C oven and bake for 30 minutes, or until the pastry is deeply golden and crisp.
Let the tart cool for 5 to 10 minutes. Run a knife around the edge to loosen the pastry, then carefully invert the tart onto a large serving plate. Top with burrata and drizzle with balsamic glaze. Serve warm or at room temperature.