In a saucepan over medium heat, combine the cream, half-and-half, and vanilla bean, stirring occasionally to make sure the mixture doesn't scorch on the bottom. When the cream mixture reaches a fast simmer (do not let it boil), set aside to infuse for 10 to 15 minutes.

In a medium bowl, whisk together the egg yolks and sugar. Whisking constantly, slowly pour the still-hot cream mixture into the egg yolk mixture. Return the mixture to the saucepan and 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, the mixture is not quite thick enough yet. When it is ready, quickly remove it from the heat and whisk in the green tea. Meanwhile, half-fill a large bowl with ice water. Strain the mixture into a smaller bowl to smooth it and remove the vanilla bean. Rest the smaller bowl in the ice water and let the mixture cool, stirring often, then continue according to the directions of your ice cream maker.