Whisk granulated sugar, salt, and cornstarch together in a medium sized saucepan. Set aside.
Whisk egg yolks, half & half, and key lime juice together in a separate bowl.
Over medium heat, whisk the liquid into the saucepan.
Bring the mixture to a boil over medium heat, whisking constantly.
Once boiling, cook the mixture for 30 seconds to 1 minute (whisking constantly), until the custard has thickened and coats the back of a spoon (mine took roughly 45 seconds, then it was done).
Remove from heat and add unsalted butter, key lime zest, and vanilla extract. Whisk until combined.
Transfer the custard to a bowl. Place the bowl with the custard in a larger bowl that has been filled with ice. Whisk the custard to cool it down.* (See notes for an alternative method.)
Once cool, pour the custard over the crust and smooth it into an even layer. Set aside.