Butter a rimmed sheet pan or line it with a silicone baking mat. Measure the vanilla and baking soda into separate small bowls and set them by the stove, because the last steps go fast.
Combine the sugar, corn syrup, water, and salt in a heavy 3-quart saucepan. Stir over medium heat until the sugar dissolves and the mixture starts to boil, then clip on a candy thermometer.
Boil without stirring until the syrup reaches 240°F (soft-ball stage), about 8 to 10 minutes.
Stir in the peanuts and butter. Keep cooking, stirring constantly, until the thermometer reads 300°F (hard-crack stage) and the syrup is a deep golden amber, about 10 to 12 minutes more.
Remove the pan from the heat. Quickly stir in the vanilla, then the baking soda. The candy will foam up, so stir just until blended.
Immediately pour it onto the prepared pan and spread it as thin as you can with a buttered spatula. Work quickly, because it sets fast.
Let the brittle cool completely, about 45 minutes, then break it into pieces. Store airtight at room temperature.
