Prepare the baking dish by lining it with parchment paper or greasing it well. Set aside.
In a medium saucepan, combine granulated sugar, butter, milk, and salt.
Place the saucepan over medium heat and stir continuously until the mixture comes to a boil.
Once boiling, stop stirring and allow it to boil undisturbed for about 5 minutes, or until it reaches the soft ball stage (about 234°F or 112°C).
Remove the saucepan from heat and add in the lemon zest, lemon juice, and vanilla extract. Stir well to combine.
Gradually add in the powdered sugar, one cup at a time, stirring until smooth and well blended.
Pour the mixture into the prepared baking dish and spread it evenly with a rubber spatula.
Allow the fudge to cool at room temperature for about 30 minutes, then refrigerate for at least 2 hours until set.
Once set, cut the fudge into small squares and serve.