Preheat the oven to 200ºC (400ºF) conventional. Grease a baking tray with a little oil.
Place the carrots onto the prepared tray, transfer to the oven and roast for 30-40 minutes, until soft. Set aside to cool.
Reduce the oven to 160ºC (320ºF). Grease and like a 23 cm (9 in) cake tin with baking paper, then grease again and dust with flour, tipping out the excess.
Transfer the roasted carrot to a food processor and pulse until the carrot is broken up into small pieces but not completely puréed. Set aside.
Place the flours in the bowl of a stand mixer with the whisk attached, then pour in the melted butter or margarine and mix on low speed for 30 seconds until the mixture is a sandy consistency.
Place both the sugars, baking powder, and salt in a large bowl and whisk to break up the brown sugar. Add to the flour mixture along with the walnuts, and mix together for about 30 seconds until the sugar has blended into the flour.
Add the milk, roasted carrot mixture, vanilla bean paste, ground cardamon, and lemon extract to a large jug and stir to combine. Pour into the flour and sugar mixture and beat on the lowest speed for 20-30 seconds, until just combined. The mixture will be very lumpy and runny.
Transfer the batter to the prepared tin.
Bake on the middle rack of your oven for 47-50 minutes, until a skewer inserted into the centre of the cake comes out almost clean. The top should be firm to touch.
Allow the cake to cool in the tin for 15 minutes, then turn out onto a cake rack to cool completely.