Place milk in a medium saucepan over medium-high heat. Heat to 180° (check temperature with an instant-read thermometer or candy thermometer) or until tiny bubbles form around edge of pan (do not boil). Meanwhile, place egg yolks in a large bowl; stir with a whisk. Once milk is heated, gradually add half of hot milk to yolks, stirring constantly to temper. Return yolk-milk mixture to saucepan .
Combine sugar, cream, and butter in a large saucepan over medium heat; bring to a boil, stirring until sugar melts. Cook 3 minutes without stirring. Remove from heat; stir in sea salt. Gradually and carefully add caramel mixture to yolk mixture, stirring constantly. Return pan to low heat; cook until a thermometer registers 160°. Place pan in a large ice-filled bowl until completely cooled, stirring occasionally.
Pour mixture into the freezer bucket of an ice-cream freezer; freeze according to manufacturer's instructions. Once ice cream is at desired consistency, serve or freeze for a few hours for a firmer consistency. Scoop about 1/2 cup ice cream into each of 10 dishes; sprinkle evenly with flake salt.