Generously oil an 18-inch-square marble slab (or an inverted baking sheet) and a thin metal spatula. Sift the baking soda and salt onto a small sheet of waxed paper. Measure the vanilla extract into a small container. Set all of these near your work area, along with a pair of rubber gloves.
In a 4-qt., deep, heavy-based saucepan, combine the water, corn syrup, and sugar. Stir over medium-low heat until the sugar dissolves, 10 to 12 minutes. When the solution is clear and begins to boil, increase the heat to high and stop stirring. Put a candy thermometer in the solution, holding it with a mitt to protect your hand. When the mixture registers 265°F on the thermometer, 8 to 10 minutes. Later, add the nuts and stir gently to disperse them through the mixture. Continue cooking, stirring occasionally, until the mixture reaches the hard-crack stage, 305° to 310°F, about 5 minutes longer. Remove the pan from the heat. Stir in the softened butter, the vanilla extract, and then the baking soda and salt. The mixture will begin to foam.
Stir just until the mixture foams evenly, and without delay pour it onto the oiled marble slab. The mixture should spread to about 14 inches in diameter. Slip the oiled spatula under the hot candy to loosen the edges and bottom. Put on the gloves and as soon as the candy is firm enough on the bottom to be picked up (the top won't be hard yet), lift the edges and turn the entire piece of brittle over. With gloved hands, stretch the brittle to extend it so it's as thin as you can get it, about 17 inches in diameter. Let the candy cool undisturbed for at least 1 hour and then break it into small pieces. Store the brittle in airtight containers for up to 10 days.
An oiled surface makes the candy easier to handle. Use marble or an inverted baking sheet -- the candy is hot, over 300°F, when you pour it out.