Preheat the oven to 200C/180C Fan/Gas 6.
Melt half of the butter in a 19cm/7½in ovenproof frying pan. Add the beetroot, sugar, vinegar and thyme leaves and cook for 12 minutes, stirring. Remove the pan from the heat and carefully cover with kitchen foil.
Bake for 1 hour, or until the beetroot are just tender. Remove the foil, dot with the remaining butter and leave to cool for at least 10 minutes – don’t forget that the pan handle will be extremely hot, so take care to only move with an oven cloth.
Increase the oven temperature to 220C/200C Fan/Gas 7.
Unroll the pastry sheet and place a 24cm/9½in dinner plate gently on top. Cut around the plate to give a 24cm/9½in pastry disc. Place the pastry on top of the softened beetroot in the pan and tuck the sides down with a round-bladed knife. Prick lightly with a fork.
Bake the tart for about 25 minutes, or until the pastry is puffed up and golden-brown. Leave to cool in the tin for 5 minutes, then place a serving plate or board on top of the pan and very carefully invert so that the tarte drops gently down – take care as the pan and handle will remain hot.
Dot with small pieces of goats’ cheese and tiny mint leaves. Serve with a lightly dressed salad.