Place 1/3 cup of sugar, lime juice, lime zest, and 1 egg in a saucepan over low heat. Whisk constantly until the mixture thickens and almost reaches a simmer. Remove from heat and tint to desired green color. Set aside to cool.

Preheat oven to 325°F (165°C).

In a bowl, cream the butter with 1/4 cup of sugar. Stir in 1 egg and vanilla extract. Mix in flour and salt until dough is smooth.

Roll dough into 1-inch balls and place on an ungreased baking sheet, about 2 inches apart. Create an indentation in the center of each ball with your thumb. Spoon about 3/4 teaspoon of lime filling into the center of each indentation.

Bake until the cookies are set but not browned, about 12-15 minutes. Allow cookies to cool on baking sheet for 1 minute before transferring to wire racks.