Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a large bowl, cream together the softened butter, granulated sugar, and light brown sugar using an electric mixer on medium speed until the mixture is light and fluffy, about 2-3 minutes.
Add the egg and vanilla extract to the butter and sugar mixture. Beat until fully combined.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined.
Stir in the rolled oats by hand using a spatula until evenly distributed.
Using a tablespoon or small cookie scoop, drop rounded spoonfuls of dough onto the prepared baking sheets, spacing them about 2 inches apart. Flatten each cookie slightly with the back of the spoon to encourage thinness.
Bake in the preheated oven for 10-12 minutes, or until the edges are golden brown and the centers look set but still soft.
Remove the baking sheets from the oven and allow the cookies to cool on the sheets for 5 minutes before transferring them to a wire rack to cool completely. This helps them crisp up as they cool.