Melt the butter in a medium saucepan with tall sides over medium heat. Stir in the brown sugar and kosher salt. Bring to a boil, stirring occasionally, about 3 minutes. (The mixture will go from looking like wet sand to thick brownie batter when stirred.)
Once boiling, carefully whisk in the cream. The sugar may clump a bit at first, but keep stirring until it is smooth again. Reduce the heat to medium-low and add a candy thermometer to the pot if you have one.
Boil, stirring frequently (make sure you get the sides and corners of the pan), until it reaches 220°F for a thinner sauce or 225°F for a thicker sauce or until it falls off a spoon in a sheet, 1 to 3 minutes.
Remove from the heat and stir in the vanilla. Let the sauce cool for 10 minutes, then transfer it to an air-tight container and let it cool completely. The sauce will thicken as it cools. Sprinkle with flake salt when serving, if desired. Store the sauce in the refrigerator for up to 2 weeks. Reheat it before serving.
Notes
Use a heavy-bottomed pan. This will ensure heat is distributed evenly so the sauce cooks evenly. A pot with a thin bottom can burn the sugar easily, especially in the first stage of cooking.
You can use light brown sugar. This will yield a lighter-flavored and lighter-colored sauce.
Skip the corn syrup. Some recipes call for corn syrup, but I don’t think it is a necessary ingredient for a silky sauce.
Use room temperature heavy cream. If the heavy cream is too cold, it could curdle due to the fast and extreme temperature change of pouring cold cream into very hot sugar. Warmer or room temperature cream will result in a silky smooth sauce.
Don’t overcook it. You will end up with a sauce that tastes bitter and burnt.
Consistency tips: A thinner sauce is perfect for using cold or drizzling over cold desserts like ice cream. Cook the sauce for a little longer to make a thicker sauce to pour onto warm or room temperature desserts like cakes or brownies.