Warm the milk, sugar, and salt in a medium saucepan. Make sure the sugar and salt completely dissolve.
Pour the cream into a metal bowl set in a larger bowl of ice and set a medium-mesh sieve on top.
Temper egg yolks: In a separate bowl, whisk together the egg yolks. Slowly pour the warm milk into the egg yolks, whisking constantly, then scrape the warmed egg yolks back into the saucepan.
Heat custard until thick: Stir the mixture constantly over medium heat with a wooden or heatproof rubber spatula, scraping the bottom as you stir, until the mixture thickens and coats the spoon, about 5-7 minutes.
Pour the custard through the strainer and stir it into the cream. Then stir until cool over the ice bath. Chill mixture thoroughly in the refrigerator.
Add peppermint extract: One the mixture is thoroughly chilled, add peppermint extract, a 1/4 teaspoon at a time, tasting the mixture after each addition, until you reach the desired level of pepperminty-ness. (Different peppermint extracts vary in strength. I used 2 teaspoons of McCormick peppermint extract, which was just the right amount for our taste.)
Process in ice cream maker: Once chilled, process in your ice cream maker according to the manufacturer's instructions.
Fold in crush peppermint candy: Once the ice cream has been formed in the ice cream maker, it will be fairly soft. Fold in the crushed peppermint candy. Put in an airtight plastic container and place in the freezer for at least an hour, preferably several hours. If it has been frozen for more than a day, you may need to let it sit at room temperature for a few minutes to soften it before serving.
