In a small saucepan, boil the carrot slices in water until very tender. Drain them then make a puree and reserve it.
In a saucepan over medium heat, heat the half-and-half, cream, vanilla, orange rind, and cinnamon stick, whisking occasionally to make sure the mixture doesn't burn or stick to the bottom of the pan. When the cream mixture reaches a fast simmer (do not let it boil), turn off the heat and let the flavors infuse for 10 minutes.
Whisk the egg yolks and sugar together. In a thin stream, whisk half of the cream mixture into the egg yolk mixture. Then pour the egg-cream mixture back into the saucepan containing the rest of the cream mixture. Heat over medium heat, stirring constantly with a wooden spoon. At 160 degrees F, the mixture will give off a puff of steam. When the mixture reaches 180 degrees F it will be thickened and creamy, like eggnog. If you do not have a thermometer, test it by dipping a wooden spoon into the mixture. Run your finger down the back of the spoon. If the stripe remains clear, the mixture is ready; if the edges blur, the mixture is not quite thick enough yet. When it is ready, quickly remove it from the heat.
Meanwhile, in a bowl put 2 handfuls of ice cubes in the bottom, and add cold water to cover. Rest a smaller bowl in the ice water. Combine the carrot puree with the custard, mixing well. Pour the cream mixture through a fine sieve or chinois (to remove the vanilla bean pieces, orange rind, and cinnamon sticks) and into the smaller bowl. Chill 3 hours, then continue according to the directions of your ice cream maker.