Line a large, rimmed baking sheet with parchment paper. In a small, heavy saucepan, combine the sugar, salt and 3/4 cup water and place over medium-high heat. Stir briefly with a wooden spoon, then attach a candy thermometer to the side of the pan and bring the syrup to a boil. Continue boiling, without stirring, over medium-high heat until the mixture is light amber and the thermometer registers 340 degrees , 12 to 15 minutes.
Working carefully and quickly, remove the thermometer and stir in the peanuts. Immediately pour the mixture onto the prepared baking sheet. Using a silicone spatula or wooden spoon, spread the mixture so the peanuts are in a single, even layer. Let cool completely, about 20 minutes. Using your hands, break the peanut brittle into pieces.