Heat the oven to 180°C/Fan 160°C/Gas 4.
Make the sponge. Mix together the ground almonds and flour. Beat the butter and sugar in a stand mixer fitted with the beater, on medium speed for 10 minutes, until pale and creamy. Add the eggs, one at a time, beating well between each addition. If the mixture starts to curdle, add a spoonful of the flour mixture. Stir in the buttermilk. Fold in the flour mixture, vanilla and salt.
Divide the mixture equally between the prepared tins and spread it level. Bake the sponges on the middle shelves for 30–35 minutes, until risen and a skewer inserted into the centres comes out clean. Leave to cool slightly in the tins.
While the sponges are baking, make the peach and prosecco compôte. Mix the peaches, lemon juice, sugar and prosecco in a pan and bring the mixture to the boil. Reduce the heat to low and simmer for 30 minutes, stirring occasionally, until the compôte thickens and the peaches are soft. Remove from the heat and leave for 10 minutes, then purée with a hand blender. Leave to cool until needed.
Make the prosecco syrup. Warm the prosecco and sugar in a small pan over a low heat until the sugar dissolves.
Using a wooden skewer or cocktail stick, make holes over the top of the warm sponges and carefully brush over the syrup. Leave the sponges in the tins for 15 minutes to cool.
Make the Chantilly cream. Whip the cream, honey and vanilla together in a bowl to reach soft peaks. Cover and chill until needed.
To assemble, place one sponge on a cake board and spread a layer of cream on top. Spoon a ring of cream around the edge of the sponge and fill the centre with the compôte. Place the second sponge on top and repeat with the cream and compôte. Add the third sponge and top with a generous dollop of cream, spreading it to the edge. Decorate with peach slices, drizzle over any remaining prosecco syrup, then finish with mint sprigs and dust with icing sugar.