Prepare a ice bath by filling a large bowl 3/4 of the way full with ice and cold water. Combine rice milk, sugar, and salt in a medium saucepan and bring to a simmer over medium heat, whisking until sugar is dissolved.

Whisk egg yolks in a large bowl until smooth. Slowly pour the warmed milk mixture into the yolks, a little at a time, whisking constantly. Scrape mixture back into the saucepan. Clean and dry the bowl then set the bowl in the ice bath and place a fine mesh strainer on top.

Place milk mixture over medium-low heat and cook until mixture thickens slightly, stirring constantly with a wooden spoon or rubber spatula, about 10 minutes. Make sure you are scraping the sides and bottom of the saucepan frequently to prevent the eggs from curdling. The custard won't get super thick like a dairy-based custard but will still leave a trail when you draw your finger through the custard on the back of the spoon.

Pour mixture through the strainer into the bowl set over the ice bath. Stir in the peppermint extract and cool stirring occasionally. Cover and transfer to the refrigerator to chill completely, at least 4 hours or overnight.

Freeze custard in the ice cream maker according to manufacturer's instructions. The ice cream will still be pretty soft; transfer to a freezer-safe container, and fold in the crushed candy. Let ice cream freeze for at least 6 hours (this may take shorter or  longer depending on your ice cream maker and freezer). While the ice cream is freezing the candy will fall to the bottom so give it a stir a few times during freezing to distribute the candy evenly.