Place milk in a medium saucepan over medium-high heat. Heat to 180° or until tiny bubbles form around edge of pan (do not boil). Place egg yolks in a large bowl; stir with a whisk. Gradually add half of hot milk to yolks, stirring constantly. Return yolk mixture to pan.

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 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 can of an ice-cream freezer; freeze according to manufacturer's instructions. Drain ice water from freezer bucket; repack with salt and ice. Cover with kitchen towels, and let stand 1 hour or until firm. Scoop about ½ cup ice cream into each of 10 dishes; sprinkle evenly with flake salt.