Combine the sugar, corn syrup, water, and salt in a medium to large saucepan. Bring to a simmer over medium-high heat. No need to stir.
Brush the sides of the pot down with a clean and oil-free pastry brush (preferably one only used for water or sugar) to make sure all the grains of sugar are dissolved.
Cover the pot for 5 minutes to let the condensation dissolve any remaining sugar crystals and help prevent crystallization.
Remove the lid and attach a candy thermometer if using one.
Continue cooking the sugar until it's a deep golden amber color or the candy thermometer reads 350º (hard crack stage). This can take from 5-10 minutes so keep an eye on it to avoid burning.
Remove from the heat. Slowly whisk in a couple Tablespoons of the cream. It will bubble up and steam, but that's normal. Continue whisking in the remaining cream in a slow stream.
Add the butter, salt, and vanilla. Stir until smooth and creamy.
Let the sauce cool. It will thicken as it cools down.
Notes
Before you begin:Have a clean, oil-free saucepan and utensils ready, and measure out the cream, butter, salt, and vanilla so they're within reach once the sugar starts boiling.Ingredient notes:Use a completely clean pot and utensils for this recipe. Any trace of grease or leftover sugar crystals can cause the caramel to seize or crystallize. Corn syrup helps keep the sugar from recrystallizing as it cooks, but you can swap in glucose syrup if that's what you have on hand.Pan options: A heavy-bottomed saucepan works best since it distributes heat evenly and helps prevent scorching. If you double the recipe, use a bigger pot. The syrup bubbles up quite a bit once the cream goes in, and you don't want it boiling over.Make-ahead and storage: Make it up to 1 month ahead and keep it in the refrigerator, or freeze it for up to 3 months. To reheat, microwave in 5 to 7 second bursts, stirring between each burst. Overheating in one long burst can cause the sauce to separate.Substitutions: Swap in brown sugar for a richer, more molasses-forward caramel, though the texture will be a little thicker. No corn syrup on hand? Glucose syrup works the same way, or leave it out if you're making a dry caramel instead of this wet-method version. Half-and-half can replace the heavy cream in a pinch, though the sauce won't be quite as thick.Critical do-nots:Don't stir the sugar syrup once it starts boiling. Agitating it is one of the fastest ways to end up with grainy, crystallized caramel. Don't walk away once the syrup starts to color. It can go from deep amber to burnt in less than a minute. Don't pour finished caramel onto parchment paper. It sticks, so reach for a heat-safe jar or container instead.