Place the basil, mincemeat, cherries and angelica in a large bowl and mix well to combine.
In the bowl of a stand mixer fitted with the paddle attachment, beat the ice cream until just softened (be careful not to over-beat as it will melt). Add the mincemeat mixture to the ice-cream bowl and stir to evenly distribute.
Lay out three pieces of cling film on a work surface, side by side but overlapping each other by half. Spoon the ice-cream mixture into the middle and use the cling film to roll it into a 7.5cm/3in thick cylinder. Place in the freezer for 24 hours.
Warm the sugar syrup and 50ml/2fl oz of the dark rum in a small saucepan. Prick the sponge cake all over with a fork. Pour the rum mixture evenly over the top of the sponge and leave to soak in.
Preheat the oven to 230C/210C Fan/Gas 8 and line a baking tray with baking paper.
In the clean, grease-free bowl of a stand mixer fitted with the whisk attachment, whisk the egg whites until soft peaks form. Then, whisking constantly, add the sugar one tablespoon at a time, whisking until stiff and glossy.
Place the soaked sponge cake on the baking tray. Remove the cling film from the ice cream, then lay the ice cream cylinder in a spiral shape on top of the sponge. Quickly spoon over the meringue using a large palette knife, spreading it thickly and in peaks over the ice cream and the sides of the cake right down to the baking tray.
To make the compôte, put all the ingredients in a saucepan and simmer for 15 minutes.
Bake for 4–5 minutes, or until golden brown (or alternatively, you can use a chef’s blowtorch to caramelise the meringue all over). Warm the remaining rum in a small saucepan, pour it over the Alaska and carefully set light to it. Make sure the flames have died down completely before serving with the compôte.