(The caramel can be made a day ahead and softened again in the microwave for easy pouring. It really doesn't take a lot of time though.)
Pour the sugar into a medium saucepan and heat to medium-high. Using a rubber spatula, stir the sugar just occasionally, scraping the bottom of the pan. At first the sugar will be lumpy, but will soon melt and start turning brown. Don't stir too often, just enough to keep everything evenly cooking. After about 5 minutes, the sugar should be fully melted and become a rich amber color.
At this point, remove the pan from the heat and immediately add the butter. the mixture will start to bubble quite rapidly. That's just what you want.
Using your spatula or a whisk, stir the butter and sugar until they melt together almost completely. This can take 1-2 minutes.
Next, pour in the heavy whipping cream and sea salt. If you're making regular caramel, then add ½ teaspoon of salt and no more. If you're making a salted caramel version, add anywhere from 1 teaspoon to 1 ½ teaspoons depending on your liking.
Stir constantly now until the sauce is all mixed together and is smooth. Place back on the stove at medium-high heat again for about 20 seconds more. Remember to keep stirring.
Remove from heat and serve immediately for a hot option over ice cream or let cool at least 15 minutes for room temperature use.
Over several minutes, the caramel will become harder to stir. For a manageable sauce, just heat in microwave for 10-15 seconds and stir again.