Preparation of Oven and Baking Tray: Set the oven to a temperature of 350°F. Prepare a baking sheet by lining it with parchment paper.

Combining Butter and Sugar: In a medium-sized saucepan set over medium heat, melt the unsalted butter and dark brown sugar until they merge into a consistent brown paste. Minor separation in the form of streaks or puddles of butter is acceptable; they needn't amalgamate entirely.

Incorporation of Flour: Remove the saucepan from the heat source. Introduce the all-purpose flour, vanilla extract, and kosher salt into the mixture. Allow this batter, which will be notably thin, to cool for approximately 10 minutes.

Addition of Egg and Oats: Once the mixture has attained a safe temperature—warm but not scalding—integrate the egg, ensuring thorough mixing. Subsequently, incorporate the thick-cut rolled oats until they are evenly dispersed throughout the batter.

Shaping of Cookies: Utilize a tablespoon to allocate portions of the batter onto the pre-prepared baking sheet, ensuring a spacing of at least 2 inches between each portion. Employ the back of the spoon to mold each portion into a fine circle, such that the oats are uniformly spread. Should space be limited on one baking sheet, it is advisable to bake the cookies in sequential batches.

Baking Process: Place the baking sheet in the preheated oven and bake the cookies for a duration of 8 minutes. The process is complete when the peripheral edges of the cookies adopt a dark golden brown hue. Once baked, allow the cookies to cool in their entirety on the parchment paper. It should be noted that the cookies should detach effortlessly from the parchment paper post-cooling, but an offset spatula can be employed if any adherence is observed.

Storage Recommendations: To maintain optimal freshness, the cookies should be stored in an airtight container. It is advisable to layer the cookies between parchment paper within the container. Under these conditions, the cookies will retain their quality for up to 3 days.