In a medium saucepan, combine the heavy cream and whole milk. Heat over medium-low heat until it's just about to simmer. Do not boil.

In a separate large bowl, whisk together the granulated sugar and egg yolks until pale and slightly thickened. Slowly pour the hot cream mixture into the yolks, whisking constantly to prevent the eggs from curdling.

Return the mixture to the saucepan and cook over medium heat, stirring constantly with a wooden spoon until it thickens slightly and coats the back of the spoon. Do not let it boil.

Remove from heat and pour through a fine-mesh sieve into a clean bowl. Stir in the vanilla extract. Allow it to cool to room temperature.

Divide the custard base evenly into three separate bowls. Stir dark chocolate into one bowl, milk chocolate into the second, and white chocolate into the third. Stir until each chocolate fully melts and integrates.

Chill the mixtures in the refrigerator for at least 4 hours, or until thoroughly cold.

Churn each chocolate mixture separately in an ice cream maker according to the manufacturer’s instructions.

To create a marbled effect, layer each ice cream alternatively in a container and gently swirl with a knife.

Freeze the mixture in an airtight container for at least 1 hour before serving.