To make the apple purée, cook 250g peeled and chopped cooking apples with a splash of water in a pan with a lid until soft for about 10 mins. Purée with a hand blender or in the small bowl of a food processor.
Butter and line the base of a 20cm springform cake tin with parchment. Heat oven to 180C/160C fan/gas 4. Beat the butter and sugar until light and fluffy. Mix the maple syrup and egg together and gradually add them. Sift the flour with the bicarb and baking powder. Add a good pinch of salt and the nuts. With a large metal spoon, fold the flour into the butter and sugar, alternating with the apple purée. Scrape the batter into the prepared tin. Bake for 40 mins, or until a skewer inserted into the centre comes out clean. Cool in the tin, then turn out onto a rack and peel off the baking parchment.
To make the buttercream, beat the butter until soft, then add the icing sugar and beat until smooth. Gradually add the maple syrup, beating as you do so. Slice the cooled cake in half then, using a spatula, spread half of the icing over the first layer. Add the other layer on top, then top with the rest of the buttercream. Pulse the pecans in a food processor with the brown sugar (or finely chop and mix). You should end up with a coarse mixture. Scatter this over the top of the cake.