Preheat the oven to 180°C/gas mark 4. Heat the oil in a frying pan over a low heat and gently cook the onions, 1 tbsp of thyme, vinegar and beetroot for 20 minutes, stirring occasionally.
Meanwhile, lay a sheet of filo across a 20cm round loose-based tart tin and brush with some of the butter. Continue layering with filo, brushing with butter as you go. Trim the edges so they hang 1cm over the side. Prick the base with a fork and cook on a baking sheet in the oven for 10 minutes.
Spoon half of the onion and beetroot mixture into the pastry case and scatter over half the goat’s cheese. Top with the remaining onion mixture and goat's cheese. Bake in the oven for 10 minutes. Garnish with extra thyme leaves, then cut the tart into wedges and serve with the salad.