Begin by gathering everything, and I do mean everything, that you will need and place it next to the stove. This includes all of your ingredients (with butter already in pieces) as well as utensils. I usually have a couple of heat-resistant silicone spatulas and a pastry brush sitting in water for use, if needed, during cooking.

Place the sugar in a medium-sized nonstick saucepan over medium heat (very important that the heat be no higher than medium) and heat, stirring constantly with a heat-resistant spatula, for ~12 minutes until sugar completely breaks down into a dark amber-colored liquid and no longer looks grainy, being careful not to let it burn (seriously, stir nonstop. If you don't, the sugar will burn and it will be ruined).
Note #1: You'll feel like you're stirring forever with nothing happening but around the 8 minute mark it'll start forming some clumps and by 10 minutes it should be almost all clumps. Everything will go very fast from there on out. Note #2: When the mixture is all liquid, toss the spatula you've been using into the sink (it will probably have some hardened sugar on it) and quickly wipe the inside of the pan down with a wet pastry brush or wet paper towel to remove any sugar crystals that are stuck to the side (DO NOT scrape sides with spatula). If some sugar chunks from your spatula got into the liquid, keep stirring (using a new heat-resistant spatula) and they should dissolve.

Moving very quickly, add the butter to the pan and, using your fresh heat-resistant spatula, stir nonstop and vigorously (but carefully to avoid splatter as the mixture is VERY HOT) for ~2-3 minutes until the butter is completely mixed in (butter will be separated for a while, but vigorous stirring will get it to thoroughly incorporate).

As soon as the butter is thoroughly incorporated and your mixture is nice and thick, add the cream SLOWLY with one hand (the cream is colder than the caramel at this point and everything will splatter like crazy if added too quickly) while stirring quickly & constantly with the other, until everything is nice and smooth and thoroughly mixed (about a minute).

As soon as the cream is mixed in and the caramel is nice and smooth, STOP STIRRING (very important) and allow to boil for 1+1/2 minutes (exactly 1+1/2 minutes, no more or less).

Remove from heat and allow to sit for 1 minute, then pour the caramel into a heat-resistant container (DO NOT scrape edges of pan at all, just pour and let whatever is stuck on the pan stay there. Trust me, you don't want it). If you're salting your caramel, add the salt now and stir in until combined (the reason you wait to add the salt until it has been removed from the pan is because stirring in the pan runs the risk of hitting the sides of the pan and getting crystallized sugar in your caramel).

Whether or not you're using the salt, allow the caramel sauce to sit at room temperature for 3 minutes, then add in the vanilla extract and stir in to thoroughly incorporate it throughout the caramel. Note: You don't want to add the vanilla in right away or else it will all boil off and also may cause the caramel to crystalize a bit. Waiting 3 minutes lets the caramel cool just enough that it's safe to add the vanilla.

If not using the caramel sauce right away, allow it to continue to cool at room temperature until completely cooled, then store caramel in a sealed container in the refrigerator for up to 1 week.