In a large heavy saucepan, combine the butter, brown sugar, milk, and salt. Bring mixture to a boil over medium heat, stirring occasionally, until mixture comes to a full rolling boil. Without stirring, let mixture boil for 3 minutes. Immediately remove from heat.
Stir in peanut butter and vanilla, stirring until peanut butter has melted.
Quickly stir in oats, mix well.
Working quickly, drop mixture by tablespoonfuls onto prepared baking sheets. Allow to cool and become firm, about 30 minutes.