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To make the cake:.
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Preheat the oven to 350F
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Butter and flour a 12-cup muffin pan.
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Mix the carrots and lemon juice and set aside.
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Combine the dry ingredients in the large bowl of an electric mixer and blend well on low speed.
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Add the oil and blend again.
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Increase the speed to medium and add the eggs, one at a time, mixing well between each addition.
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Scrape down the sides of the bowl.
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Stop the mixer and stir in the carrot mixture.
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Spoon the batter into the prepared muffin cups, filling each not quite full.
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Place in the middle of the oven and bake 30 minutes, or until the edges of the cakes pull away from the sides of the cups and a toothpick inserted in the center of a cake comes out clean.
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Take out of the oven and cool on a cake rack for 10 minutes.
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Unmold and cool the individual cakes to room temperature on the rack.
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To make the frosting:
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Put the cream cheese and butter in the bowl of an electric mixer and beat on medium speed until combined.
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Scrape the bowl and beaters.
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With the mixer on medium, slowly add confectioners sugar.
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(Pour the sugar at the edge of the bowl, not directly into the beaters, to avoid creating a cloud)
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When all the sugar has been incorporated, beat in the vanilla.
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Put the frosting in a pastry bag fitted with a small star tip (could use sandwich bag with tip cut).
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Slice the cakes in half horizontally and fill with a thin layer of cream cheese frosting.
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Put the tops back on.
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Pipe many little stars of frosting on top of each cake.
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Set aside for at least half an hour to let the frosting harden slightly.
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To make the creme anglaise:
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Put the half-and-half in a heavy saucepan over medium heat.
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Split the vanilla bean and scrape the seeds into the cream; drop in the pods.
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Bring just to a boil.
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Remove from heat and set aside to steep for 15 minutes.
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Remove the vanilla bean pods.
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Set a medium bowl in a larger bowl of ice.
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In another medium bowl, whisk together the egg yolks and sugar until the mixture is pale yellow.
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Whisking constantly, very slowly pour the scalded cream into the egg mixture until it is all incorporated.
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Stir in the Grand Marnier.
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Pour the custard back into the saucepan and cook over medium heat, stirring slowly with a wooden spoon, until the custard thickens.
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(Do not boil!
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).
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Remove the custard from the heat and strain through a fine-mesh sieve into the bowl set in ice.
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Stir with a whisk while cooling.
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To serve: Fill the center of each dessert plate with creme anglaise.
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Place a frosted carrot cake in the center of each.
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Arrange orange segments around the cakes.
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Place a mint sprig on each.
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Dust the entire plates with confectioners sugar.
