Add the butter, sugar, milk, and cocoa powder to a medium 3-quart saucepan and heat over medium, whisking until the butter has melted. The mixture should look smooth, glossy, and deep brown before it begins bubbling.

Let the mixture come to a full rolling boil, then stop whisking and boil it for 1 minute. Start the timer only when rapid bubbles break across the center of the pan, not when small bubbles cling to the edges; an instant-read thermometer should reach 225°F (107°C).

Take the saucepan off the heat and promptly stir in the peanut butter and vanilla, followed by the oats and salt. The chocolate mixture should become thick and shiny, falling from the spatula in a heavy ribbon.

Leave the mixture in the saucepan for 5 minutes. During this pause, the oats absorb moisture and the mixture gains enough resistance to hold a rounded scoop.

Line two baking sheets with parchment paper or silicone baking mats while the oat mixture rests, leaving room in the refrigerator for both sheets. The lined surfaces prevent the warm cookies from sticking as they firm.

Portion the mixture onto the prepared sheets with a 1-Tablespoon cookie scoop or a Tablespoon, flattening the tops lightly with the back of a spoon if desired. The scoops should sit in mounds rather than spreading into thin puddles.

Refrigerate the cookies for 30-60 minutes, then remove them once they have set. They are ready when the surface looks matte and the edges feel firm enough to lift cleanly from the parchment.