Add the butter to a medium saucepan and cook over medium-low heat. As it melts, the butter will begin to foam and sizzle. Stir constantly, scraping the bottom of the pan, until small brown bits form. This should take about 6–8 minutes. Remove from heat immediately.

Stir the brown sugar and granulated sugar directly into the warm browned butter until smooth and fully combined.

Transfer the mixture to the refrigerator and chill for about 10 minutes to cool slightly.

While the mixture chills, preheat the oven to 325°F. Line two baking sheets with parchment paper and set aside.

Remove the cooled butter mixture from the refrigerator. Stir in the eggs one at a time, followed by the vanilla extract, mixing until smooth.

Add the flour, oats, baking soda, and salt to the bowl. Mix just until the dough comes together and no dry flour remains.

Gently stir in the white chocolate chips and semi-sweet chocolate chips by hand until evenly distributed.

Scoop the dough using a 2-tablespoon cookie scoop and place onto the prepared baking sheets, leaving about 2 inches between each cookie. Lightly press each one down.

Bake for 11–13 minutes, until the edges are lightly golden but the centers still look soft. Avoid over baking for the best texture.

Let the cookies cool on the baking sheets for a few minutes, then transfer to a wire rack to cool completely.