Place an 10 inch ovenproof pan over medium heat. Add the butter and sugar to the pan, then gently swirl the pan until the sugar is dissolved. Add the thyme leaves and season with salt andpepper.
Place the shallots in the pan, cut side down, packing them in until they completely cover the base of the pan then turn the heat down to low and allow to caramelize for 10 minutes or until the shallots are soft.
Pour in the balsamic vinegar, swirl the pan to distribute and continue to simmer for 2 minutes then remove from the heat. Using a small sharp knife, cut a circle a little larger than the size of the pan out of the puff pastry. Cover the shallots with the pastry, ensuring the you tuck the edges in around the walls of the pan (this will help to hold all the juices in when you turn out the tart).
Bake the tart at 400°F (200°C) for 20 minutes or until the pastry is puffed up and golden. Set aside for 5 minutes to cool slightly. Gently turn the tarte tatin by placing your serving dish over the top of the pastry and inverting the pan. Garnish with crumbled blue cheese and slice. Serve warm. Cover and store in the fridge for up to 2 days.