Line an 8x8 pan with foil or parchment paper, then grease sides that are not covered by the paper or foil, and set aside (you can also grease the entire pan which helps the parchment paper stick)
Set a medium sized heatproof bowl over a wire rack, and add 1 tablespoon of butter and vanilla extract. Set aside
Set medium sized heavy-bottom pot over the stove and add sugar, heavy cream, corn syrup, and salt.
Insert a candy or digital thermometer to monitor the temperature (see NOTE 1 below).
Heat mixture over medium-low heat, and stir to combine until the sugar has melted. During the first 5 minutes or so, wash down the sides of the pot with a damp pastry brush to prevent sugar crystals from forming.
You can opt to grease the sides of the pot instead of using the pastry brush to wash the sides.
Once the mixture is homogenous and the sugar has melted, stop stirring and remove the spoon from the mixture! Otherwise, you can end up with grainy fudge.
Raise heat until the mixture reaches a very gentle simmer.
Continue to heat without stirring (no spoon!) until the mixture reaches 238F, aka the soft ball stage.
If you don't have a thermometer, use a bowl of ice cold water and once your fudge looks like bubbling lava, drop a teaspoon full into the cold water. Remove the fudge from the water, and squeeze it between your fingers - If it balls up into a soft ball, your fudge is ready.
When the mixture reaches the target temperature, immediately transfer the contents to the heatproof bowl containing the butter and vanilla. Do not use a spoon to transfer and do not scrape sides or bottom of pot.
Allow the mixture to cool undisturbed until it reaches 110F - do NOT stir while it is cooling down.
After the mixtures has cooled (about 1.5 hours), begin stirring with a wooden spoon until it thickens and loses its gloss (10 to 15 minutes). The more continuous and plentiful the stirring, the more fine grained and smooth your fudge will be (yes, I take a few breaks!)
Immediately transfer to prepared pan using a wooden spoon or spatula, smooth top, and let it rest on the counter for at least another hour before refrigerating.
Refrigerate for at least 8 hours or until set, remove paper or foil, and cut into 64 square pieces.