To make the marzipan, heat the sugar and 75ml/5 tbsp water in a saucepan until dissolved then boil the syrup until the mixture reaches 100C/220F (use a sugar thermometer to measure this). This is known as the soft ball stage, when a drop placed in cold water to cool down, it will form a soft ball.
In a bowl, mix together the ground almonds, egg white and almond extract.
Pour the sugar syrup over the ground almonds and mix vigorously until the mixture forms a thick paste.
Chill the marzipan in the fridge for 1-2 hours, or until firm.
For the tart, preheat the oven to 210C/425F/Gas 7. Line a baking tray with baking parchment.
Roll the puff pastry out thinly on a floured surface. Cut out 4 x 12cm/4½ in circles, and lay them onto the baking tray. Leaving a gap of about a centimetre round the edge, lightly score the middles of each pastry circle with a small sharp knife, making a grid of lines. This will stop the pastry puffing up too much in the middle. Brush each pastry circle with the beaten egg.
Roll the chilled marzipan out to 5mm/¼in thick. (If the marzipan is too hard to rollout, allow it to warm slightly at room temperature.) Cut out 4 x 11cm/4in circles and place each marzipan circle on top of a pastry circle.
Arrange the apple slices, overlapping in concentric circles, on top of each circle of marzipan so that all of the marzipan is covered.
Heat the butter in a saucepan until just melted then brush over the top of the apples and sprinkle the sugar over the top.
Transfer to the oven to bake for 15-18 minutes, or until the pastry is risen and golden-brown.
Meanwhile, place the apricot jam and a tablespoon of water into a saucepan and heat until the jam has melted. Pass the jam through a fine sieve into a bowl.
Remove the tarts from the oven and brush with the melted apricot jam.
For the sauce, heat the sugar and butter in a small saucepan until melted and well combined.
Pour in the double cream and simmer gently for 3-5 minutes, or until the mixture thickens slightly.
To serve, place a tart into the centre of each serving plate, drizzle the caramel sauce around the outside. Top with a ball of ice cream and sprig of mint and serve immediately.