Step 1: Prepare your workstation. Thoroughly butter a large, heat-proof surface like a marble slab, a large baking sheet, or a silicone mat. Also, butter your hands generously, as you'll be pulling warm candy.
Step 2: Combine sugar, heavy cream, water, and salt in a heavy-bottomed 2-quart saucepan. Stir gently over medium heat until the sugar dissolves. Avoid stirring once the sugar has dissolved to prevent crystallization.
Step 3: Increase heat to medium-high and bring the mixture to a boil. Insert a candy thermometer, making sure it doesn't touch the bottom of the pan. Cook without stirring until the mixture reaches 245°F (118°C) – the firm ball stage. This process can take 15-20 minutes, so be patient.
Step 4: Once the candy reaches 245°F (118°C), immediately remove the pan from the heat. Carefully add the cubed butter and vanilla extract, but do not stir. Allow the butter to melt into the hot candy for about 2-3 minutes without disturbing it.
Step 5: Gently pour the hot candy mixture onto your prepared buttered surface. Let it cool undisturbed until it's warm enough to handle, but still pliable – typically about 15-20 minutes. Test by touching the edge; it should be warm but not scalding.
Step 6: Now for the fun part: the pulling! Generously butter your hands again. Gather the candy into a manageable rope. Begin to pull the candy, stretching it out and then folding it back onto itself. Repeat this process, continuously pulling and folding. As you pull, the candy will become opaque, lighter in color, and develop a beautiful silky sheen. This typically takes 10-15 minutes, or until the candy is firm enough to hold its shape but still pliable.
Step 7: Once the candy is pulled to the desired consistency and color, roll it into a long rope about 1/2 to 3/4 inch thick. Using buttered scissors or a sharp, buttered knife, cut the rope into 1-inch pieces. Wrap each piece individually in wax paper or parchment paper to prevent sticking. Allow them to cool completely before storing.