Pour oats into a large bowl and set aside. Line a baking sheet with parchment paper. Note: A baking sheet isn’t “required” – a flat surface, like a countertop, works.
Melt 1 stick butter in a saucepan on medium-high heat, stirring frequently to melt butter. Bring to a boil for 1 minute, then remove from heat. Add peanut butter and vanilla extract and stir to combine.
Pour peanut butter mixture over reserved oats. Add milk and sugar, and stir until fully incorporated.
Scoop 1 tablespoon full of batter onto baking sheet, pressing down to flatten.
Melt chocolate chips with 1 tablespoon unsalted butter in a microwave safe dish, 20 second intervals on medium heat, stirring after each interval, until chocolate is completely melted.
Use a spoon to drizzle chocolate over top of cookies. Cookies are ready to eat once chocolate has cooled and set. Or, chill in refrigerator for 1 hour. Transfer cookies to a serving platter, or store in an airtight container.