Place 1/3 cup of sugar, the lime juice, lime zest, and 1 egg in a saucepan over low heat, and whisk constantly until the mixture almost reaches a simmer and begins to thicken. Remove the mixture from the heat, and tint to your desired shade of green with food coloring. Set the filling aside to cool.

Heat oven to 325 degrees F (165 degrees C).

In a bowl, cream the butter with 1/4 cup of sugar, then stir in 1 egg and vanilla extract. Mix in the flour and salt until the dough is smooth. Roll dough into 1-inch balls. Arrange on an ungreased baking sheet, placing the balls about 2 inches apart. Use your thumb to make an indentation in the center of each dough ball. Spoon about 3/4 teaspoon of the lime filling into each.

Bake in the preheated oven until the cookies are set but not browned, 12 to 15 minutes. Allow cookies to cool on baking sheet for 1 minute before removing to finish cooling on wire racks. When cookies are cool, sprinkle lightly with confectioners’ sugar.