Preheat your oven to 375°F and line your baking sheets with parchment paper.
In a small saucepan over medium heat, melt the butter until it turns golden brown and gives off a lovely nutty aroma. Take it off the heat and let it cool for about 5 minutes.
In a mixing bowl, mix together the brown sugar, oats, egg, and vanilla. Slowly pour in the cooled brown butter and stir until everything is well combined.
Add in the salt and flour, mixing gently until you have a nice dough.
Using a cookie scoop, drop dollops of the mixture onto the lined baking sheets, making sure to leave enough space between each cookie for spreading.
Bake in the preheated oven for around 6 minutes or until they're beautifully golden brown.
Once baked, let the cookies cool on the baking sheet for about 5 minutes before transferring them to a wire rack to cool completely.