Bring half and half and whipping cream to a simmer over medium-low heat in a medium-heavy saucepan.
In a medium sized bowl, whisk egg yolks and sugar together until smooth.
Once cream mixture starts simmering, whisk hot cream into egg yolk mixture in a slow, steady drizzle. Return mixture to saucepan and stir continuously over medium-low heat until mixture thickens and coats the back of a spoon. Do not boil.
Remove from heat and stir in Black Raspberry Jam or Sour Cherry Jam until melted.
Cover with plastic wrap, making sure the plastic wrap sits directly on top of the custard and place in refrigerator to chill. Freeze in an ice cream maker according to manufacturer's directions.