Mix the whole milk and 1 cup of the sugar in a medium saucepan. Heat on medium stirring frequently until it reaches 170 degrees on an instant-read thermometer.
In a separate medium mixing bowl, whisk the egg yolks and the remaining 1/4 cup sugar together. Whisk until the mixture is foamy and slightly thickened.
Slowly temper the egg yolk mixture by drizzling about half the hot milk into the eggs while whisking continuously. Whisk the tempered egg mixture into the remaining whole milk and sugar mixture in the saucepan. Return the saucepan to medium heat. Cook, stirring continuously with a wooden spoon, until the mixture registers 185 degrees on an instant-read thermometer or when it is thick enough to coat the back of the spoon. Do not allow the mixture to boil or you'll make scrambled eggs. Remove from the heat.
Pour the heavy cream into a large clean glass or metal bowl, set over an ice bath. Pour the heated milk and egg mixture through a fine sieve into the cream. Add the peppermint extract and stir until well blended. Stir the mixture occasionally until it has cooled completely (this should take about 30 minutes.) Remove the bowl from the ice bath, cover and refrigerate overnight or at least 8 hours.
Process the chilled mixture in your ice cream maker according to manufacturer instructions. Remove the finished ice cream to a large bowl and mix in the 1/2 cup of crushed peppermint. Store in an airtight plastic container and freeze for several hours before serving. Garnish with additional peppermint if desired.
Makes 1 1/2 quarts.