Pre-heat the oven to 375°F/190°C
Melt the butter and oil over medium heat in an oven proof frying pan. Add the sliced beets and red onions toss to coat in the butter and oil. Add the balsamic vinegar, brown sugar, rosemary, thyme, salt and pepper and stir well, covering the beets and onion.
Arrange the beetroot and onions with the cut side facing down in the frying pan, filling in any spaces nice and snug. Cover with foil and place into a preheated oven for 30 minutes until beets and onions are soft and caramelized in the balsamic sherry glaze.
Cut the Puff Pastry the size of the frying pan and cover the beetroot and onions, tucking in the edges of the pastry in the pan. Bake for 25 minutes, until pastry is golden brown.
Allow the Tarte Tatin to cool for about 10 minutes. Place a large plate over the pan, and quickly flip it over, removing the pan.
Crumble Goats Cheese over the Tarte Tatin, and sprinkle fresh thyme and rosemary.