In a heavy-bottomed saucepan, combine the heavy cream, whole milk, half of the granulated sugar (3/8 cup), and salt. Heat over medium heat, stirring occasionally, until the mixture is hot and just barely simmering around the edges, but not boiling. Remove from heat.

In a separate large bowl, whisk the egg yolks with the remaining granulated sugar (3/8 cup) until light and creamy.

Temper the egg yolks: Slowly pour about half of the hot cream mixture into the egg yolk mixture, whisking constantly to prevent the eggs from scrambling. Once combined, pour the egg yolk mixture back into the saucepan with the remaining hot cream.

Return the saucepan to medium-low heat. Cook, stirring constantly with a spatula, until the custard thickens slightly and coats the back of a spoon (it should reach about 175-180°F or 80-82°C). Do not boil. Immediately remove from heat.

Strain the custard through a fine-mesh sieve into a clean bowl. Stir in the vanilla extract and peppermint extract. If using, add a few drops of red gel food coloring and stir until desired festive pink color is achieved.

Cover the bowl with plastic wrap, pressing it directly onto the surface of the custard to prevent a skin from forming. Refrigerate for at least 4 hours, or preferably overnight, until thoroughly chilled.

Once the base is completely chilled, churn it in your ice cream maker according to the manufacturer's instructions. This usually takes about 20-30 minutes, until the ice cream is the consistency of soft-serve.

In the last 5 minutes of churning, add the crushed peppermint candies. Churn until just combined.

Transfer the soft ice cream to an airtight freezer-safe container. Press plastic wrap directly onto the surface and freeze for an additional 2-4 hours, or until firm enough to scoop.

Serve immediately and enjoy your homemade Christmas Peppermint Ice Cream!