Preheat oven to 350°F and line a baking sheet with parchment paper.
In a medium saucepan over medium heat, melt the butter and brown sugar into a thick paste, stirring frequently.
Remove from heat and stir in flour, vanilla, and salt. Let cool 10 minutes.
Add egg and oats and mix with a fork until well combined.
Spoon out onto prepared baking sheet in 1 tablespoon portions 2 inches apart. Using the back of a spoon, spread each cookie into a thin circle so oats are in a single layer.
Bake until edges begin to turn golden brown, about 8 minutes. Remove from oven and let cool completely on parchment paper.
Once cool, cookies should peel easily from parchment. Enjoy!
Note: To store, layer between pieces of parchment paper in an air-tight container.