Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together the all-purpose flour, rolled oats, 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 just until combined. Do not overmix.
Using a tablespoon or small cookie scoop, drop dough onto the prepared baking sheets, spacing about 2 inches apart. Flatten each cookie slightly with the back of a spoon or your fingers to encourage thinness.
Bake for 10 to 12 minutes, or until the edges are golden brown and the centers look set but still soft.
Remove from the oven and allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. This helps them crisp up.
Store cookies in an airtight container at room temperature for up to 5 days.