Make the sugared peaches. Combine the sliced peaches and ¼ cup sugar in a small saucepan. Bring to a simmer over medium-low heat and simmer for 3 minutes. Turn off the heat and mash with a potato masher or fork until it is the consistency of chunky applesauce.
Warm the milk, cream, and some of the sugar. Add the milk, cream, and ½ cup of the sugar to a large saucepan. Over medium heat, warm the mixture until it is very warm (175℉), about 5 minutes.
Beat the eggs and remaining sugar. While the mixture is heating, combine the egg yolks with the remaining ¼ cup of sugar. Beat until it turns a lighter yellow in color.
Temper the eggs. 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, heat over medium, and cook. Stir constantly and cook until the mixture is 185℉, or has thickened enough to coat the back of a spoon. Do not boil.
Chill the custard. Remove the custard from the 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.
Churn and add peaches. 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 peach mixture.
Freeze to set. Cover and refreeze 2-4 hours until hardened. Scoop and serve. It will last up to a week in the freezer.