Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together the rolled oats, all-purpose flour, baking soda, and salt until well combined.
In a large bowl, whisk the melted butter, granulated sugar, and light brown sugar until smooth and combined.
Add the egg and vanilla extract to the butter and sugar mixture and whisk until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, stirring gently with a spatula until just combined. Do not overmix.
Scoop about 1 tablespoon of dough per cookie onto the prepared baking sheets, spacing them about 2 inches apart. Flatten each dough ball slightly with the back of a spoon or your fingers to encourage thinness.
Bake in the preheated oven for 10 to 12 minutes, or until the edges are golden brown and the centers look set but still slightly soft.
Remove the cookies from the oven and allow them to cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely. This helps them crisp up further.
Store cooled cookies in an airtight container at room temperature.