Line one or two baking sheets with parchment paper or wax paper. Make sure the sheets are level so the cookies can cool into evenly shaped rounds.

Add the sugar, unsalted butter, and milk to a saucepan over medium heat. Stir frequently as the butter melts and the ingredients combine.

Bring the mixture to a genuine full rolling boil, then cook for about 1 minute while stirring. Begin timing only after the mixture is actively bubbling across the surface.

Remove the saucepan from the heat and quickly stir in the peanut butter and vanilla extract. Continue mixing until the peanut butter melts into a smooth, glossy mixture.

Fold in the quick-cooking oats until every oat is evenly coated. Work promptly because the mixture will begin thickening as it cools.

Drop tablespoon-sized portions onto the prepared paper. Leave a little space between cookies, then let them stand at room temperature for 20 to 30 minutes, or until firm.