Add the milk, cream, and 1/2 cup of the sugar to a large saucepan. Over medium heat, warm the mixture until it is very warm (175 degrees), about 5 minutes.
While the mixture is heating, combine the egg yolks with the remaining cup of sugar. Beat until it turns lighter yellow in color.
Remove the milk mixture from the heat, and half a cup at a time, add roughly half of it to the egg mixture. Mix after each addition. This keeps it from cooking the eggs. Once you have blended half of the milk into the eggs, pour the entire milk- egg mixture back into the saucepan. Place it all back on the heat and heat over medium and cook. Stir constantly and cook until the mixture is 185 degrees, or has thickened enough the coat the back of a spoon. Do not boil.
Remove from heat and stir in the vanilla and salt. Strain the mixture using a stainer or cheesecloth set over a large bowl. Allow to cool at room temperature, then cover and place in fridge until very cold- at least 6 hours, up to overnight.
Remove custard from fridge and set up ice cream maker, according to manufacturer's instructions. Churn until it is the consistency of soft serve ice cream, then add the crushed peppermints 1/4 cup at a time.
Cover and refreeze 2-4 hours until hardened. Scoop and serve. Will last up to week in the freezer.