Cut each plum half into three slices. Butter a 20 cm springform tin and line the base with greaseproof paper.
Put the butter, sugar,eggs, lemon zest, flour and nuts into a food processor and whizz until smooth. Spoon the batter into the tin and arrange the plum slices, cut side up, in circles on top.Pack them well together.
Sprinkle with 1 tbsp caster sugar then bake in an oven preheated to 180 C, 160 Fan, Gas 4 for 40 minutes. A skewer inserted into the centre should come out clean.
Let the cake cool in its tin for 15 minutes, then remove it, peeling off the greaseproof paper. Put on a wire rack to cool completely.