In a medium saucepan, combine lemon juice, 1/4 cup sugar, the corn syrup, and butter. Cook over medium heat, stirring occasionally, until syrup reaches a boil, 5 to 6 minutes. Remove from heat and set aside.
In a medium bowl with a mixer, beat eggs, egg yolk, and remaining 1/2 cup sugar until slightly frothy. Beating on low speed, slowly drizzle warm lemon syrup over egg mixture until well blended. Transfer mixture back to saucepan and cook over low heat, stirring constantly, until curd thickens enough to coat the back of a metal spoon in a thin, even layer (nappe), 4 to 5 minutes. Remove from heat and let cool.
Drizzle lemon curd over ice cream and sprinkle with mint and chives.
Make ahead: The lemon curd, chilled airtight up to 2 weeks.