1¼ cups unsalted butter, cut up
4 large eggs, at room temperature
1 cup granulated sugar
1 tablespoon lime zest
1 tablespoon fresh lime juice
1 tablespoon vanilla extract
½ teaspoon coarse kosher salt
2 cups all-purpose flour
½ teaspoon baking powder
2 tablespoons unsalted butter, melted and cooled
1 cup powdered sugar
2 tablespoons lime zest, divided
2 tablespoons fresh lime juice
1 to 2 tablespoons whole milk