Add egg yolks to a medium-sized mixing bowl along with half of the sugar. Whisk vigorously or beat with a hand mixer until the mixture becomes pale and slightly thickened, approximately 2-3 minutes. Set aside. *See Note #3 below.
In a saucepan over medium heat, combine the heavy cream, whole milk, and remaining sugar. Stir until the mixture is warm and the sugar is dissolved, about 3-4 minutes. Be careful not to boil the mixture.
Add the crushed peppermint candies to the warm cream mixture. Stir until the candies are mostly melted and the mixture is infused with peppermint flavor. Remove from heat.
Temper the egg yolk mixture by slowly pouring in 1/2-1 cup of the warm peppermint milk mixture into the egg yolks, whisking constantly. *See Note #4 below.
Pour the egg and milk mixture back into the saucepan. Cook over low heat, stirring constantly, until the mixture thickens and coats the back of a spoon, about 8-10 minutes. Do not let it boil.
Remove the custard from heat and stir in the peppermint extract and vanilla extract. If you want a pink hue, add a few drops of red food coloring and mix until well incorporated.
Refrigerate the mixture, covered, for a minimum of 4 hours, preferably overnight.
Stir in the liqueur to the chill ice cream base. Pour the chilled custard into an ice cream maker and churn according to the manufacturer's instructions until it reaches a soft-serve consistency.
Transfer the churned ice cream into a lidded container, layering it with extra crushed peppermint candies if desired. Freeze for several hours or until firm.