Make the cake batter: Combine flour, granulated sugar, ground pistachios, baking powder, baking soda, salt and spices in a medium bowl and whisk to blend evenly.
Combine carrots, brown sugar, oil, thinned out yogurt, egg and vanilla in another medium bowl and whisk until combined. Make a well in the center of the dry ingredients and add the wet mixture. Fold until evenly combined.
Spread evenly into the pan and bake for 20-25 minutes. Cool for 20 minutes, then transfer to a cooling rack.
Make the frosting: Beat soft cream cheese in a large bowl using an electric hand mixer until very smooth. Add 2 Tbsp of cream and beat until there are no lumps. Then gradually add remaining cream a few tablespoons at a time and beat well between additions until it forms soft peaks. Gradually beat in sugar 1 Tbsp at a time until it is thickened to a spreadable consistency, then mix in lemon juice and vanilla extract.
Spread the frosting over the cooled cake and then top with more crushed and chopped pistachios before serving. The frosting will be soft at first, but it will thicken as it sets and after it is refrigerated. CS
