1/3 cup water
1/2 cup sugar — + 2 tablespoons
1/4 cup very finely minced — (packed) pureed or grated
1 cup whole milk
2 extra large egg yolks
1 cup cold heavy cream
1 teaspoon fresh lemon juice
Combine the water and 1/4 cup of the sugar in a small heavy saucepan. Bring to a steaming near-simmer over moderate heat, stirring to dissolve the sugar. Add the ginger, stir to blend, then bring the mixture to a boil. Reduce the heat to maintain a weak simmer and simmer for 5 minutes. Remove from heat.
Combine the 2 tablespoons sugar with the milk in a large heavy saucepan. Bring to a near simmer over moderate heat, stirring to dissolve the sugar. Remove from heat. Carefully scrape the sugar syrup into the milk mixture. Stir to combine, cover the pot to hold in the heat, and set the mixture aside to steep for 20 minutes.