Line two baking sheets with parchment paper or wax paper and set them aside. Measure the oats and vanilla before cooking so they can be added immediately when the syrup is ready.

Add the unsalted butter, whole milk, and granulated sugar to a saucepan over medium heat. Stir as the butter melts and continue heating until the mixture reaches a full rolling boil.

Once the mixture reaches a vigorous rolling boil, start a timer and boil for about 1 minute while stirring continuously. Do not substantially exceed this time because overcooking can make the cookies dry and crumbly.

Remove the saucepan from the heat and immediately stir in the pure vanilla extract. Adding the vanilla off the heat helps preserve its fragrant flavor.

Add the quick-cooking rolled oats and stir until every oat is evenly coated with the hot vanilla syrup. Work promptly while the mixture is warm and easy to scoop.

Drop heaping tablespoonfuls of the mixture onto the prepared parchment or wax paper. Leave room between the cookies, then let them stand at room temperature until firm, usually 20–30 minutes.