Melt the butter in a microwave-safe bowl. Let it cool for about 5 minutes. Hot butter will melt the sugar and cause greasy cookies. Once cooled to room temperature, whisk or use an electric hand mixer to mix in the brown sugar until fully combined. Add the egg and vanilla and whisk until smooth and creamy.
Add the baking soda, baking powder, and salt and whisk until smooth. Add the properly measured cake flour and all-purpose flour, mixing just until combined. The dough will be thick, but should not be dry. Do not overmix. Gently fold in the chocolate chips and walnuts.
Use a 1/2 cup measuring cup to divide the dough evenly to form 5 large dough balls. Place on a parchment paper lined pan in the fridge for 1 hour.
Preheat the oven to 325°F. Line a sheet pan with a silicone baking mat and do not use foil. I have found parchment is too thin and can cause the bottoms to brown too quickly. Place 3 dough balls on the pan at a time, spacing them well apart. Bake for 13-20 minutes, erring on the side of underbaking for soft, chewy cookies. The cookies will continue to set as they cool. Visual cues: The tops should look set, not wet or gooey.
Remove the cookies from the oven and work quickly to gently press the edges of each cookie inward using the back of a metal spatula. If desired, press a few extra chocolate chips or walnuts onto the tops. Let the cookies cool on the pan for 10 full minutes, then transfer to a cooling rack with a metal spatula. If you like a salty-sweet finish, sprinkle a few flakes of salt on top while the chocolate is still soft.
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Notes
Note 1: Cake flour gives these cookies a softer, more tender crumb. Using both cake flour and all-purpose flour creates the bakery-style texture. Do not substitute or use homemade cake flour. It will not work the same. Too much flour leads to cakey/dry cookies. Don't pack flour into the measuring cup and make sure to level off the top. If you have a food scale, swap to grams and use those measurements for accuracy.Note 2: I recommend toasting the walnuts first. Let them cool, then coarsely chop.Storage: These cookies are best fresh but will keep up to 1 week in an airtight container at room temperature. Rewarm in short 10 to 15 second bursts.Freezing: Freeze the dough, not the baked cookies. Freeze dough balls until solid, then store in an airtight container for up to 3 months. Bake from frozen, adding a few extra minutes until edges are lightly browned and centers are soft.