In a medium saucepan, whisk together the sugar, cornstarch, and salt.
Gradually whisk in the milk and heavy cream until smooth.
Cook over medium heat, stirring constantly with a whisk, until the mixture thickens and comes to a simmer (about 5-7 minutes).
Remove from heat.
In a separate bowl, whisk the egg yolks.
Temper the egg yolks by slowly whisking in about 1/2 cup of the hot milk mixture. Whisk constantly.
Pour the egg yolk mixture back into the saucepan and whisk to combine.
Cook over low heat, stirring constantly, until the custard thickens enough to coat the back of a spoon (about 1-2 minutes). Do not boil.
Remove from heat and stir in the lemon juice, lemon zest, and vanilla extract.
Pour the custard into a bowl and press plastic wrap directly onto the surface to prevent a skin from forming.
Refrigerate for at least 2 hours, or until completely chilled.