For the pastry, sieve the flour and sugar into a large mixing bowl. Tip the cubed butter into the bowl. Rub the butter and the flour between your thumb and fingertips until the mixture resembles fine breadcrumbs.
In a jug or small bowl, beat the egg together with four tablespoons of ice-cold water. Pour into the flour mixture.
Slowly bring the ingredients together with your hands to form a dough, being careful not to overwork it.
Knead the dough lightly on a clean, floured work surface, then wrap it in cling film and refrigerate until firm, at least 30 minutes.
For the frangipane, beat the butter and sugar together in a bowl until light and fluffy. Crack the eggs into the bowl one at a time, beating well after each addition. Add the ground almonds and mix well until combined. Set aside.
Preheat the oven to 180C/350F/Gas 4. Dust the work surface with flour and roll the chilled dough out thinly. Use it to line one large 25cm/10in tart ring or 6-8 individual tartlet rings 8cm/3¼in in diameter. Trim away any excess.
Spoon the frangipane into the tart case so that it comes about halfway up the sides. Smooth over the surface with a spatula and cover the frangipane evenly with the plums.
Bake for 30-40 minutes (15-25 minutes for the tartlets), or until the pastry is crisp and golden-brown and the fruit is tender.
Remove the tart(s) from the oven. Dust with icing sugar and serve warm with whipped cream or crème fraîche.