Gather and measure all ingredients.
Line 2 large cookie sheets with parchment or wax paper. My countertops are heat-resistant, so I line them instead. Less cleanup.
Place the butter, sugar, and milk in a saucepan and heat over medium heat on the stovetop. Stir often, making sure that the mixture melts and blends and does not stick. Bring the mixture to a rolling boil. It should be bubbling vigorously. Once you see nothing but rolling bubbles, continue stirring and quickly set a timer for 60 seconds.
When the timer goes off, remove from the heat and stir in the peanut butter and the vanilla extract. Mix well until all the ingredients are blended well. Immediately add the quick-cook oats and stir until the oats are covered with peanut butter mixture.
Drop spoonfuls of the mixture onto the lined cookie sheets or countertops. I will use a tablespoon to scoop each cookie, then slide the cookie off with a teaspoon and spread each cookie out a little as I go along. Remember the mixture is still hot. You will need to do this process as quickly as possible before the mixture firms up.
Allow the cookies to cool for about 45-60 minutes to completely firm up. If the cookies have not firmed up after an hour, let them continue to sit out a while longer, and they will firm up.
Once cookies are cool and have firmed up, carefully peel them from the parchment or wax paper and store them in an air-tight container. Cookies can last in an air-tight container at room temperature for about a week.