Preheat the oven to 200°C, fan 180°C, gas 6. In a 26cm ovenproof frying pan, melt the butter and sugar over a medium heat. Cook for 5 minutes to soften and caramelize the beets. Finish with a sprinkling of chopped rosemary.
Roll out the pastry to the thickness of a 1 coin on a lightly floured surface. Cut a circle 2cm larger than the frying pan's diameter.
Spread the horseradish sauce on one side of the puff pastry. Place the beetroot in an even layer in the pan, cut-side down, and then place the pastry circle sauce-side down over the beetroot, tucking the edges under the pan's side.
Brush with beaten egg before baking for 35-40 minutes, or until golden and puffy. Allow it cool for 5 minutes before carefully running a knife down the pan's edge, placing a large serving dish on top, and inverting the pan to flip out the tart. To serve, top with crumbled goat's cheese and thyme leaves. If desired, serve with additional horseradish sauce.