Butter a rimmed baking sheet; set aside. Stir together sugar, 1/2 cup water, and the salt in a medium saucepan. Cook over high heat, without stirring, until sugar begins to melt and turn golden, about 3 minutes. Continue to cook, stirring occasionally, until sugar has melted and mixture turns golden amber, about 10 minutes.
Remove pan from heat. Stir in peanuts. Immediately pour peanut mixture onto buttered baking sheet. Quickly spread mixture to 1/2 inch thick using an oiled metal spatula. Let cool completely, about 15 minutes. Break brittle into pieces.
Use a medium-size saucepan (wider rather than taller) that does not have a nonstick coating.