Place milk, cream, 1/4 cup sugar, ginger, and ginger root in a small pan and cook over low heat, whisking now and then, until warm (175F). Set aside for 1 hour, then strain out the solids. Keep 2 tsp of the ginger root.
Whisk egg yolks, 1/4 cup sugar, vanilla extract and salt in a small metal bowl until combined. Temper with the milk mixture, then add to the saucepan and cook until thickened, about 185F. Do not boil. Strain out the solids.
Cool for 3 hours.
Add reserved ginger root back in. Prepare in ice cream maker. Add candied ginger in the last 5 minutes of processing.