Cut butter into pieces and melt in a light-colored saucepan over medium heat, swirling occasionally. Cook 4 to 6 minutes until the foam subsides, the milk solids turn deep amber, and the butter smells like hazelnuts or toffee. Immediately pour all the butter and amber milk solid specks into a large heatproof bowl.
Add both sugars to the hot browned butter. Whisk vigorously 60 to 90 seconds until incorporated and smooth. Let cool until warm to the touch, 5 to 8 minutes.
Add the whole egg, extra yolk, and vanilla. Whisk for 60 seconds until smooth and glossy. Rest 30 seconds, whisk again 30 seconds. Repeat this cycle a total of 3 times over 5 minutes.
Add flour, baking soda, and fine salt. Fold with a spatula using broad strokes until no flour streaks remain; stop immediately once incorporated.
Fold in the chocolate chips until evenly distributed. Preheat oven to 190°C (375°F).
Scoop 3-tablespoon portions onto parchment-lined baking sheets with 6cm spacing. Press 2 to 3 extra chips on top of each. Add a small pinch of flaky salt to each.
Bake one tray at a time on the center rack for 10 to 12 minutes until edges are golden-brown and set but centers still look slightly underdone and matte.
Remove from oven. Immediately bang the tray once firmly on the counter to deflate the centers. Rest on the tray 8 to 10 minutes before transferring. Centers will set fully during this time.
