Measure the ingredients, line large baking sheets with parchment paper, and keep them within reach before heating the saucepan. The mixture thickens fast after the oats are added, so the prepared pans should be ready before the syrup boils.

Combine the butter, sugar, and milk in a large saucepan over medium heat, stirring often until the butter melts and the mixture looks smooth. Wait for strong, active bubbling across the full surface, not just small bubbles around the edges.

Once the mixture reaches a full rolling boil, stop stirring and boil it for 60 seconds. Use a timer, because starting too soon leaves excess moisture behind and boiling longer can make the cookies dry.

Take the saucepan off the heat and stir in the peanut butter and vanilla until the mixture is fully smooth. The syrup will thicken noticeably and smell strongly of warm peanut butter.

Fold in the quick-cooking oats until every oat is coated, then scoop mounds onto the parchment-lined baking sheets and gently flatten them if desired. The oats should look glossy while warm, then turn matte as the cookies cool and firm.

Let the cookies cool for 45 minutes to 1 hour before serving. They should feel set on the outside and continue to firm as they stand.