Coat a large, rimmed baking sheet with butter and set it aside. (Test Kitchen Tip: Instead of butter, you can use a silicone baking sheet, which we find offers more even coverage of the pan.)
In a medium, heavy-bottomed saucepan over medium-high heat, bring the sugar, corn syrup, water, and salt to a boil, stirring continuously until the sugar has dissolved, about 5 minutes.
Continue cooking, swirling the pan occasionally, until the mixture reaches the hard-ball stage (see Kitchen Science below) on a candy thermometer (250-265 degrees), about 5 minutes. With a pastry brush dipped in water or a heat-resistant silicone spatula, wash down the sides of the pan to keep sugar crystals from forming (all adult jobs).
Carefully stir in the nuts (the syrup will be very hot!). Continue cooking, stirring frequently to prevent the nuts from burning, until the mixture is light amber in color and the nuts are golden brown, about 5 minutes.
Remove the pan from the heat. Add the butter and vanilla extract and stir until the butter is melted. Carefully stir in the baking soda until combined (the mixture will foam up in the pan), then quickly pour the mixture onto the prepared baking sheet (an adult's job).
Using an offset or a heat-resistant spatula coated with oil, quickly but carefully spread the candy into a thin, even layer about 1/2 inch thick. Slide the spatula under the hot candy to loosen the edges and bottom.
Let the brittle cool completely, about 60 minutes. Break it into large pieces and store it at room temperature in an airtight container layered between pieces of waxed paper. It will keep for up to a month.