Preheat oven to 350°F. Grease and flour one 8-inch round cake pan. Line bottoms with parchment for easy removal.
In a medium bowl, whisk together flour, baking powder, baking soda, spices, and salt.
In a large bowl, whisk together brown sugar, granulated sugar, and oil. Add eggs one at a time, then stir in vanilla.
Add dry ingredients to the wet and mix just until combined. Fold in grated carrots and nuts.
Pour the batter into the pans. Bake for 30–35 minutes, until a toothpick inserted in the center comes out clean. Let cool in pan for 10 minutes, then transfer to a wire rack to cool completely.
In a medium saucepan, combine the cream, milk, half the sugar, cinnamon, nutmeg, salt, and the vanilla extract. Heat over medium heat until it’s steaming, just below a simmer. Remove from heat and let steep for 10–15 minutes to let the spices infuse.
Meanwhile, in a medium bowl, whisk the egg yolks with the remaining sugar until pale and slightly thickened.
Slowly pour the warm milk mixture into the yolks, whisking constantly, to temper the eggs. Once combined, pour the whole mixture back into the saucepan.
Cook over medium-low heat, stirring constantly with a wooden spoon or rubber spatula, until the custard thickens enough to coat the back of the spoon. Do not let it boil.
Strain the custard through a fine-mesh sieve into a clean bowl to remove the vanilla pod and any bits of cooked egg. Let cool slightly, then cover and chill in the fridge for at least 4 hours.
Once chilled, churn the custard in your ice cream maker according to the manufacturer’s instructions (I prefer the Kitchen Aid's ice cream attachment), about 20–25 minutes, until thick and creamy.
Cut the carrot cake into 1 inch cubes. Stir the cubes of cake into the churned ice cream.
In a medium bowl, beat the softened cream cheese and butter together until smooth and creamy. Gradually add powdered sugar, mixing well after each addition. Stir in vanilla extract and salt. Add milk or cream 1 tablespoon at a time until the icing reaches a loose, pourable consistency.
Transfer the ice cream to a lidded container one scoop at a time, drizzle the cream cheese icing on top of each layer. Freeze for at least 2 hours for a firmer scoop.