Prep the oven and pan. While the butter melts, position a rack in the center of the oven and preheat the oven to 350°F. Generously butter the bottom and sides of a 10-inch cast-iron skillet.
Mix the dry ingredients. In a small bowl, whisk together the flour, baking soda, baking powder, and kosher salt.
Mix the butter, sugars, egg, and extract. In a medium bowl, use a rubber spatula to mix together the butter and sugars until moistened, about 1 minute. Add the egg and vanilla extract and mix until combined.
Add the dry ingredients and chocolate. Gradually mix in the dry ingredients until just combined. Add the chocolate all at once and mix until the chocolate is evenly distributed throughout, about 1 minute.
Assemble the cookie dough in the pan. Transfer the cookie dough into the prepared skillet by scraping a small portion of the dough into the pan and using a wooden spoon, stiff rubber spatula, or your hands to press the dough down into the skillet. Press the dough evenly over the bottom of the pan.
Bake the cookie cake. Bake the chocolate chip cookie cake for 20 to 22 minutes, or until the edges are set but the center is still gooey. Cool on a wire rack completely before frosting. The cookie cake can be stored by slicing it into wedges after frosting. Store the wedges in an airtight container at room temperature for up to 3 days.