In a saucepan over medium heat, heat the milk, heavy cream, and vanilla, stirring 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, turn off the heat and let sit 10 minutes to infuse the flavor. Meanwhile, whisk together the egg yolks and sugar. In a thin stream, whisk the cream mixture into the egg yolk mixture. Then pour the egg-cream mixture back into the saucepan. Heat 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 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. Whisk in the mascarpone.
Meanwhile, in a bowl, put in 2 handfuls of ice cubes and add cold water to cover. Rest a smaller bowl in the ice water. Pour the cream mixture through a fine sieve or chinois (to remove the vanilla bean pieces) and into the smaller bowl. Chill until cool, about 1 hour, then stir in the crystallized ginger and continue according to the directions of your ice cream maker to freeze.