Preheat your oven to 170C/150C fan and line a baking tray with non-stick greaseproof (baking) paper.
First make the frangipane, if you’re using it. Beat the butter and sugar in a bowl until they come together. Then, add the egg and beat until combined. Finally, add the almonds and teaspoon of sugar and gently mix until it’s all a consistent mix.
Roll out your puff pastry (if it’s not pre-rolled) into a big rectangle – it should be about half a centimetre thick. Use a pizza cutter or knife to cut it into 12 smaller rectangles.
Prepare a chopping board and knife, melt a wee knob of butter in a cup in the microwave and place some icing sugar in a dry sieve or tea strainer. One by one, chop the apples into quarters, core them, and then chop each quarter into lots of thin slices. Aim for about 1-2mm thick.
As you’re chopping you can prepare each wee tart – smear a rough teaspoon of frangipane on each pastry rectangle, followed by a cascade of apple slices. Then (this will stop them going brown, too) you can brush them with some melted butter and sprinkle generously with icing sugar.
Once they are all prepared, bake in the oven for a good 30-40 minutes, until they are a very dark brown and the apples are gloriously caramelised.