In a small saucepan, bring the half-and-half and heavy cream to a gentle simmer over medium heat. Do not boil.
Turn off the heat, and add the white chocolate. Stir until fully melted and smooth.
Remove the saucepan from the heat, and add the butter, salt, and peppermint extract. Stir until everything is fully melted and incorporated.
Pour the sauce into a heatproof, airtight container.
Use right away, or let cool completely, cover, and store in the refrigerator for up to 2 weeks.