Place the pitted cherries in the bowl of a food processor. Pulse several times until the cherries are finely chopped. Set aside.
In a large bowl beat together the lemon juice and granulated sugar until well combined. While beating, slowly pour in first the milk, and then the half-and-half. Stir in the chopped cherries and juice, and the white chocolate.
Pour the mixture into the bowl of an ice cream maker. Process the sherbet according to the manufacturer's instructions, freezing until the mixture is firm but scoopable. It is best the day it's made but may be transferred to a sealed container and stored in the freezer for about 1 week.
Makes about 2 pints.
Notes
*Adapted from a recipe in the 1942 edition of The Good Housekeeping Cook Book