Preheat oven to 350 degrees. Spray a 10-inch cast iron skillet with non-stick spray.
Make the brown butter: Cut the butter in small pieces and heat the butter in a small pan on medium heat for about 3-5 minutes until the butter turns an amber color. The butter will first foam and start to pop. Then, brown bits will begin to form in the bottom of the pan. Once browned and popping slows, it is ready. You should smell a nutty aroma. Scrap the butter (and all the of those magical brown bits into a heat-safe bowl and let rest for about 15 minutes.
Whisk together the brown butter, sugar, egg, egg yolk, molasses, and vanilla extract until well combined.
In a separate bowl, sift the all-purpose flour, bread flour, baking soda, baking powder, and salt into a bowl and whisk together until combined. Then, pour the flour mixture into the wet batter and mix until incorporated. Be sure not to overmix.
Add the chocolate chips and chocolate chunks to the mixing bowl and fold into the dough.
Transfer the cookie dough to the prepared skillet. Use a rubber spatula or the back of a spoon to spread out the dough and press it into even layer in the skillet. Sprinkle extra semi-sweet chocolate chips and dark chocolate chunks on top and use a spatula to press them into the top of the dough just a bit.
Bake the cookie skillet for 17-22 minutes at 350 degrees until the edges are golden brown, the center is set, and a skewer inserted in the middle comes out clean.
Top with flaky sea salt and let the cookie skillet cool for 10 minutes before serving. Serve with vanilla ice cream, my easy homemade caramel. Add extra chocolate shavings on top if you like!