Preheat oven to 400 degrees Fahrenheit. This is not a typo! It should be 400 degrees. Line two sheet pans with parchment paper or silicone baking mats.
In a small saucepan, add the butter. Melt the butter over medium heat. Once the butter starts to sizzle, stir the butter until it turns a golden brown color and the milk solids at the bottom of the pan turn brown. Be careful not to burn it.
Transfer the butter to a large bowl to cool so it doesn't continue to brown. Add the sugar and brown sugar and stir to combine.
Add the vanilla extract, egg, and egg yolk to the mixture. Stir until incorporated.
Add the all-purpose flour, oats, baking soda, and kosher salt to the mixture. Use a spatula to stir until a cookie dough forms. The dough will be sticky!
Add the semisweet and milk chocolate chips to the dough and stir to mix in. This recipe makes 12 big cookies or 21 small cookies.
Use a large cookie scoop to scoop 6 cookie dough balls onto each pan, giving them space to spread.
Bake for 9 minutes for big cookies and 7 minutes for smaller cookies. They don't take long to bake! They will look golden brown but underbaked when they come out of the oven. They set up as they cool!
Let the cookies set up on the sheet pan for at least 10 minutes before transferring to a wire rack to cool.