Preheat the oven to 350?F and line an 8x8-inch square metal baking pan with parchment paper, leaving a 2-inch overhang on each side, and secure the sides with binder clips.

In a medium bowl, combine flour, granulated sugar, ground pistachios, baking powder, baking soda, salt, and spices, and whisk to blend evenly.

In another medium bowl, combine grated carrots, brown sugar, oil, thinned yogurt or sour cream, egg, and vanilla, and whisk until combined.

Make a well in the center of the dry ingredients and add the wet mixture. Gently fold until evenly combined; the batter will be glossy but thick.

Spread the batter evenly into the prepared pan and bake for 20-25 minutes, or until a skewer inserted into the center comes out clean. Let cool in the pan for 20 minutes, then transfer to a cooling rack.

In a large bowl, beat the soft cream cheese until very smooth. Add 2 tablespoons of cream and beat until smooth with no lumps.

Gradually add the remaining cream a few tablespoons at a time, beating well between additions until it forms soft peaks.

Gradually beat in sugar 1 tablespoon at a time until it thickens to a spreadable consistency, then mix in lemon juice and vanilla extract.

Spread the frosting over the cooled cake and top with crushed and chopped pistachios before serving.