Preheat the oven to 180 Celsius / 350 Fahrenheit (electric) or 160C / 320F fan assisted. Line the bottom of a 20 cm springform or loose bottom baking tin with a circle of parchment paper. Grease the sides with butter.
Using an electric handheld mixer or a food processor, beat the eggs with sweetener until pale and fluffy, about 2 minutes. Add the melted butter and yoghurt (or cream cheese). Mix till combined.
Mix all remaining dry ingredients together in a clean bowl. Add the dry ingredients to the wet and mix to combine. (The batter should be thick, but not dry. If your batter looks too dry, add 1-3 tablespoons of almond milk to loosen it.)
Last, stir in the grated carrots.
Spoon the cake batter into your baking pan and smooth the top with a spatula. Bake for about 55-60 minutes or until you can insert and remove a skewer without dough sticking. Check the cake after 45 minutes. If the top is golden brown, loosely cover it with aluminium paper to prevent burning.
Allow to cool in the pan for 10 minutes, then remove from the pan and allow to fully cool on a wire rack.
Mix the frosting ingredients in a bowl, then frost the cake and top with walnuts.