Combine sugar and water in a medium heavy-bottomed saucepan. Set over medium heat and stir just until sugar dissolves and the mixture looks clear, about 2 minutes. Stop stirring once it begins to boil.

Let the mixture boil undisturbed, swirling the pan occasionally, until it turns a deep amber color like copper pennies, 8 to 10 minutes. Watch closely - it goes from perfect to burnt in seconds. Remove from heat immediately.

Add the butter pieces all at once. The mixture will bubble up violently - whisk constantly until butter is completely melted and incorporated, about 1 minute.

Slowly drizzle in the heavy cream while whisking steadily. The sauce will bubble again and may seize slightly - keep whisking until smooth. Stir in salt and vanilla extract.

Let cool for 10 minutes in the pan, then transfer to a glass jar. The sauce will thicken as it cools. Store refrigerated for up to 2 weeks; rewarm gently before serving.