In a medium saucepan, combine the butter, milk, and brown sugar over low heat. Stir occasionally until the butter has completely melted, and the sugar has dissolved into a smooth blend.

Increase the heat and bring the mixture to a rolling boil, allowing it to bubble rapidly for exactly 1 minute. This step is crucial to help the mixture set properly.

Remove the saucepan from heat and immediately add the peanut butter, vanilla extract, and oats. Stir thoroughly, ensuring the oats are well coated and the peanut butter is fully incorporated, resulting in a thick and cohesive mixture.

Using a spoon, drop rounded portions onto wax paper, spacing them apart slightly. Allow the cookies to cool and set at room temperature until they’ve firmed up.