Preheat the oven to 190°C, fan 170°C, gas 5. Grease an 18cm square cake tin and line the base with baking parchment.
Put the brown sugar and eggs in a large bowl and whisk together until thick and creamy. Slowly add the groundnut oil, whisking constantly. Stir in the flour, sultanas, coconut, spices and grated carrot. Mix well until combined, then spoon the mixture into the prepared tin.
Bake for 30 minutes, or until a skewer inserted into the centre of the cake comes out clean. Remove from the tin and let cool on a wire rack.
Meanwhile, whisk together the butter, soft cheese and light brown sugar, then stir in the lime zest and chopped walnuts. Spread evenly over the top of the cooled cake with a palette knife.