Brown the butter: Melt the butter in a small saucepan over medium heat. Once melted, whisk constantly; the butter will begin to crackle, then foam. After a few minutes, the butter will begin to turn a golden amber color. As soon as the butter turns brown and gives off a nutty aroma, 5 minutes, remove from the heat and immediately transfer the bowl of a stand mixer, scraping all the butter and brown bits from the pan. Set aside to cool for 5 minutes.
Mix the wet ingredients: In the bowl of the stand mixer fitted with a paddle attachment, mix the brown butter, brown sugar, and granulated sugar on medium speed until well combined, about 1 minute. Beat in the egg, egg yolk, vanilla, and yogurt until the mixture is smooth and creamy and similar in appearance to a caramel sauce, about 1 minute. (Alternatively, you can do this with a whisk; it will take roughly the same amount of time.)
Prepare the dry ingredients: In another large bowl, whisk the flour, baking soda, and salt until well combined. With the mixer running on low speed, slowly add the dry ingredients to the wet ingredients and mix just until combined, about 1 minute. Add the chocolate chips and mix on low speed until just incorporated into the dough (don’t overmix!)
Scoop the dough: Use a large cookie scoop or spoon to measure a ½ cup (6 ounces) ball of dough. Use your hands to roll the dough ball into a sphere, then place it in a 9-inch pie pan, a 9- or 10-inch cast iron skillet, or a 9x9-inch square baking pan. Repeat with all of the dough – you should get 7 balls; if you have a little dough left, you can make the center ball extra big! Top the dough balls with 2 more tablespoons of chocolate chips.
Chill: Place the pan in the fridge to chill for 1 hour. (Chilling the dough will allow the flavors to come together, the butter to solidify so that the dough does not spread too much, and the sugar to concentrate into a deeper butterscotch flavor. Please note that the dough needs to chill to be perfect. Have patience, because the results are worth it!)
Bake: After the dough has been in the fridge for 1 hour, place your oven rack in the middle of the oven and preheat to 350°F. Bake the cookies for 18 to 23 minutes, or until the cookie dough balls have spread into each other and are just lightly golden brown on top but still very gooey in the center. It’s VERY important to not overbake the dough, or it will be dry instead of fudgy. UNDERBAKING IS ESSENTIAL! Sprinkle flaky salt over the pan right when it comes out of the oven.
Let the pan cool for 10 minutes before scooping out servings into small bowls or sundae cups. Enjoy as-is or with a scoop of vanilla ice cream.