Add the butter to a saucepan over medium heat. Cook, stirring often, until the butter foams and develops golden brown bits on the bottom with a nutty aroma, about 5–7 minutes. Remove from heat and let cool for 15–20 minutes.
In a large bowl, whisk together the browned butter, brown sugar, and granulated sugar until smooth.
Add the eggs and vanilla extract, mixing until fully combined.
.In a separate bowl, whisk together the flour, oats, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
Chill the dough for 30 minutes before baking.
Preheat your oven to 350°F and line baking sheets with parchment paper.
Scoop the dough onto the prepared baking sheets, spacing the cookies about 2 inches apart.
Bake for 10–12 minutes, or until the edges are lightly golden and the centers still look slightly soft. Add flaky sea salt on top for a sweet-and-salty bakery-style finish.
Let the cookies cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
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