Brown the butter:In a medium saucepan, melt the butter over medium heat, swirling occasionally. Continue cooking until it foams, crackles, and smells nutty. Golden brown bits will form at the bottom (about 5–7 minutes). Immediately pour into a large mixing bowl and let cool for 10–15 minutes.
Preheat the oven to 350°F. Grease a 10-inch cast iron skillet or oven-safe skillet.
Make the dough:Once the brown butter has cooled slightly, whisk in the brown sugar and granulated sugar until smooth. Add the egg, egg yolk, and vanilla, and whisk until creamy and glossy.
Add dry ingredients:In a separate bowl, mix flour, oats, cinnamon, baking soda, baking powder, and salt. Add to the wet ingredients and stir until just combined.
Fold in chocolate chips. Reserve a small handful for the top.
Press the dough into the skillet, spreading evenly. Top with remaining chocolate chips and a sprinkle of flaky salt.
Bake for 18–22 minutes, until the edges are golden and set but the center is still soft.
Serve warm, straight from the skillet with big scoops of vanilla ice cream. You will want seconds.
