To a blender, pour in the currants and milk. Blend until very smooth, about 1 minute. Place a sieve atop a large bowl and pour the puree through it, discarding the pulp.

To a medium saucepan, set over medium low heat, pour in the heavy cream, sugar and pinch of salt; stir until the sugar has dissolved, about 2 minutes. Add the sugar/heavy cream mixture to the bowl with the milk and currant mixture and stir until thoroughly combined. Cover the bowl with plastic wrap and refrigerate until very cold, at least 4 hours or overnight.

Churn the mixture in the bowl of your ice cream maker, according to your machine's instructions. Near the end of the churn, pour in the white chocolate chips; the machine will do the mixing for you. Serve immediately for soft serve or alternatively transfer to a freezer-safe quart container and freeze overnight.