Using a stand mixer fitted with the paddle attachment (or in a large bowl with a handheld mixer), beat 1 cup butter until smooth; about 30 seconds. Scrape down sides of bowl and add ⅓ cup granulated sugar, ⅓ cup light brown sugar and 1 tablespoon + 2 tsp lime zest; beat on medium-high speed until smooth and creamy, about 1 minute.
Scrape down sides of bowl and add 1 egg yolk, 1 tablespoon lime juice, ¾ teaspoon salt, and ½ teaspoon vanilla extract. Beat to combine, scraping down sides of bowl as needed.
Add 2¼ cups flour and 1 tablespoon cornstarch and mix on low just until no pockets of dry flour remain. Scrape down sides of bowl and give dough one more mix by hand to ensure no dry flour remains.
Using a tablespoon measure, scoop and roll dough into balls (16g each) and transfer to a parchment-lined baking sheet.
Use your thumb to gently press an indent in center of each dough ball, pressing about three-quarters of the way through; mend any cracked edges.
Chill indented dough balls in fridge until hard, at least 30 minutes or up to 2 days. (Indented dough balls can be frozen for up to 3 months.)
Heat oven to 375ºF (190ºC) with rack set in middle position. Line two baking sheets with parchment paper.
Arrange 8 indented dough balls on prepared sheet and fill each with about 1 teaspoon lime curd.
Bake until edges of cookies are set and bottom edges are just starting to turn golden, 10–11 minutes. Remove from oven and let cookies cool on sheet 5 minutes then transfer to a wire rack to cool completely.