Put a large mixing bowl in the freezer to chill. In a saucepan over medium heat, bring the half-and-half, cream and vanilla bean to a simmer, stirring occasionally to make sure the mixture doesn't burn or stick to the bottom of the pan. In a bowl, whisk together the egg yolks and sugar. When the cream mixture reaches a fast simmer turn it off, do not let it boil! In a thin stream, whisk half of it into the egg yolk mixture. Then pour the egg-cream mixture into the saucepan containing the rest of the cream mixture. Cook over medium heat, stirring constantly with a wooden spoon. At 160 degrees, the mixture will give off a puff of steam. When the mixture reaches 180 degrees, it will be thickened and creamy, like eggnog. If you don't 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, it is not quite thick enough yet. When it is ready, quickly remove from the heat. Meanwhile, remove the bowl from the freezer, put 4 handfuls of ice cubes in the bottom, and add cold water to cover. Rest a smaller bowl in the ice water.

Strain the cream mixture through a fine sieve to remove the vanilla bean pieces, into a smaller bowl. Chill 3 hours, then freeze according to the directions for your ice cream maker.