In a large saucepan set over medium heat, bring the lemon juice, sugar, butter, and lemon peel just to a boil, stirring frequently until sugar is dissolved. Turn off heat.
In a large bowl, whisk eggs until lightly beaten. Very gradually, whisk the warm lemon juice mixture into the eggs. Do not add the warm mixture to the eggs all at once or the eggs could cook.
Return mixture to the saucepan and set over low heat, stirring constantly until the lemon curd has thickened enough to coat the back of a spoon. Do not boil. (You should be able to draw a line on the back of the spoon with your finger without the line disappearing.)
Transfer lemon curd to a large bowl and whisk in the cream and milk until smooth. Cover and refrigerate for several hours or overnight.
Pour the chilled lemon cream mixture into an ice cream maker and process according to the manufacturer’s instructions. Transfer prepared ice cream to a container, cover, and freeze several hours until completely frozen.
Serve using the Meyer lemon as a bowl as shown in picture. Drizzle some extra zest if desired.