Peel and slice the ginger very thinly. Cover with water, bring to a boil, and cook 2 minutes. Discard the water, but keep the ginger in the pan. Add sugar, milk and 1 cup of the heavy cream. Bring up to a simmer, then turn off heat, cover, and steep until you are happy with the intensity of ginger flavor, at least one hour.
Whisk the egg yolks in a large bowl. Add the cream, a few spoonfuls at a time, until about 1/3 of the cream had gone into the egg yolks. Pour the now-warm egg yolks into the saucepan.
Cook over low heat, stirring constantly, until the custard thickens enough to coat the back of a spoon. Pour custard through a strainer onto white chocolate in a large bowl. Discard contents of strainer. Stir until chocolate is fully melted. Add remaining 1 cup of cream. Chill over ice bath or in refrigerator until thoroughly cold.
Churn in your ice cream maker according to manufacturer's instructions. Freeze at least 2 hours before serving.