24 hours before making ice cream with an electric ice-cream maker, place the freezer bowl in the freezer. Check your ice cream maker manual for specific instructions.
In a small sauce pan, heat the half and half, peppermint extract and vanilla extract on medium-low for 15 minutes. Be sure to stir occasionally and do not let it boil.
Meanwhile, in a large mixing bowl, add the yolks and sugar. With a whisk, break the yolks and combine with the sugar.
Remove the liquid from the stovetop and temper the yolk mixture. To temper the yolks, whisk the eggs quickly while slowly adding the warm liquid (about 1/4 cup at a time) until all the liquid is combined.
Cover the bowl and place in the fridge. Let it chill for at least 2 hours, up to 24 hours.
Place the ice cream mixture into the ice cream maker and mix for 20 minutes, or until the ice cream is thickened to the desired consistency. Add the crushed peppermints at the end of the mixing. Mix for a few more minutes until the peppermints are well incorporated.
Serve immediately for soft serve, or place in a freezer safe container and freeze for a firmer texture.