Freeze Nutella: Line small tray with parchment. Scoop 5 portions of Nutella (20–24g each) onto tray. Freeze 20–30 minutes until solid.
Brown butter: In light-colored saucepan over medium heat, melt butter. Cook, stirring frequently, until foamy and amber-brown specks appear with nutty aroma, about 5–7 minutes. Pour into heatproof bowl. Cool 10–15 minutes.
Make dough: In large bowl, whisk cooled brown butter with brown sugar and granulated sugar until glossy. Add egg and vanilla, whisk until smooth and thickened.
Mix dry: In separate bowl, whisk flour, baking soda, and salt. Fold into wet mixture just until no flour streaks remain. Fold in 150g chocolate chips. Do not overmix.
Assemble: Preheat oven to 175°C (350°F). Grease 8-inch cast iron skillet with butter. Press two thirds of dough (about 360–390g) into bottom and up sides of skillet, forming bowl shape.Place frozen Nutella scoops evenly across base, pressing down slightly.Flatten remaining dough (about 210–240g) into shingles. Cover Nutella scoops completely, sealing edges firmly to base. Top with 50g chocolate chips. Bake: Place skillet in oven. Bake 35 minutes. At 20–25 minutes, check edges, tent with foil if browning too quickly. Cookie pie is done when edges deep golden brown and top set.
Rest: Remove from oven. Rest 15–20 minutes.
Serve: Scoop portions directly from skillet with large spoon or ice cream scoop.