Halve the tomatoes, and place onto a plate a little larger than the base of the pan you are using for your tarte tatin. This makes it easier to check you have enough tomatoes to cover the base of the pan.
Preheat oven to 200 C (180 C fan).
Heat the butter, sugar and balsamic vinegar on a medium heat until all the ingredients melt into a homogenous caramel sauce.
Remove from the heat. Spread the sprigs of thyme over the caramel.
Place the tomatoes skin-side down into the caramel, snug up against each other. If using different varieties, take a little time to spread out the different types around the pan.
Roll out the puff pastry (if not using ready-rolled), and cut a suitably sized circle, a little larger than the size of the base of your pan. Place the pastry over the tomatoes and tuck down around the edges. Cut four or five small slits in the pastry, to allow steam to escape during cooking.
Transfer to the oven for 25-35 minutes. From the 25 minute mark, check every 5 minutes, continuing to bake until the top of the pastry turns golden brown. Remove from the oven.
Whilst the caramel is still hot and melted, place a large plate inside the pan (one that is large enough to fit snuggly to the edges of the pan). Using oven gloves, firmly grasp pan and plate and flip both over, placing them down onto a trivet or cork mat.
Lift the pan away from the plate. With luck, all your tomatoes will come cleanly away from the base of the pan, but if one or two are left in the pan, use a spoon to gently transfer them to their rightful place on top of the tarte.
Serve straight away.