Combine lemon zest, lemon juice, water, and 1/2 cup plus 2 tablespoons sugar in a saucepan; bring to a boil. Cook and stir for 2 minutes. Remove from heat.
Strain lemon mixture into a large bowl; discard lemon zest. Refrigerate until completely chilled, 8 hours to overnight.
Stir heavy cream into the lemon mixture. Pour into an ice cream maker. Freeze according to manufacturer's instructions, about 30 minutes. Transfer to an airtight container and freeze until firm, about 2 hours.