For caramelising the apricots, you need a small ovenproof frying pan with a base that measures about 20cm/8in in diameter. Put the sugar in the pan and set it over a medium heat. Cook until the sugar first melts and then caramelises and turns golden brown. Don’t stir the sugar but swirl it around the pan every now and then.
Remove the caramel from the heat and stir in the butter with a wooden spoon. The caramel will be extremely hot so watch out for splashes and don’t dream of tasting it.
Continue stirring for 2–3 minutes as the caramel cools and thickens. It will look oily and separated to begin with, but will become smooth and toffee-like as you continue stirring. When the caramel is smooth, carefully arrange the apricots on top, cut-side down. Leave to cool for 20 minutes.
Preheat the oven to 200C/400F/Gas 6.
Unroll the puff pastry sheet on a lightly floured surface and use a rolling pin to roll it out until it is 2cm/1in wider. Place a dinner plate or 25cm/10in cake tin on the pastry and cut around it.
Gently slide the pastry on top of the apricots and push down the sides with a round-bladed knife. Prick the surface to allow steam to escape.
Bake the tarte tatin for 25 minutes until the pastry is golden-brown and the apricots are cooked. Remove the pan from the oven using an oven cloth to hold the handle of the pan – don’t forget - it will be very hot.
Leave the tart to stand for a couple of minutes to allow it to settle, then loosen the edges and place a large serving plate or board on top of the frying pan. Very carefully, but quickly, turn it over, using a folded dry tea towel to help you hold it, and allow the tarte tatin to drop gently on to the serving plate.
Serve warm with crème fraiche or ice cream.