Position a rack in the center of the oven and preheat to 350°F or 175°C. Use the 5 grams of softened butter to coat the bottom and sides of a room-temperature 6 1/2-inch cast-iron skillet. Do not preheat the skillet because a hot pan would melt the dough’s fat on contact and overbrown the bottom before the center sets. Place the prepared skillet on a heat-safe surface while you mix the dough.
Place the 60 grams of melted butter, brown sugar, and granulated sugar in a medium bowl. Whisk for 45 to 60 seconds, until the mixture looks glossy and cohesive rather than separated or oily. Add the egg yolk and vanilla, then whisk for another 30 seconds until slightly thickened and smooth. The butter mixture should be warm but not hot; aim for 32 to 35°C before adding the yolk so the emulsion remains stable.
Add the flour, baking soda, and salt to the butter mixture. Fold with a flexible spatula just until only a few flour streaks remain, then fold in 40 grams of the chocolate chips until no dry pockets are visible. The dough should be soft, cohesive, and between 21 and 24°C. If it is warmer than 24°C or looks shiny with loose butter, refrigerate it for 5 to 10 minutes before shaping to limit excess spread.
Press about two-thirds of the dough evenly into the prepared skillet, using your fingertips or the back of a spoon to form a level base. Scatter the chopped chocolate over the dough, keeping it about 1/2 inch from the skillet wall so it cannot leak and scorch. Flatten pieces of the remaining dough between your fingers, arrange them over the chocolate, and gently seal the joins. Press the reserved 10 grams of chocolate chips into the top without burying them.
Bake the skillet cookie for 15 to 18 minutes. It is ready when the outer 1 inch is raised and golden, the center surface is softly puffed and matte, and no area of dough looks wet or fluid. The enclosed chocolate will remain molten, so judge the surrounding dough rather than that filling. Transfer the hot skillet to a wire rack and cool for 20 minutes. Use oven mitts because cast iron stays hot, then cut directly in the skillet into four wedges and serve warm.