Make the Brown Butter: Melt two sticks of unsalted butter in a small saucepan over medium low heat, stirring occasionally. Once the butter is melted and begins to bubble and foam, stir continuously to ensure none of the milk solids (those little brown bits that appear as the butter melts) settle to the bottom of the pan. Wait for the color to change. Lower the heat if necessary, and wait for the butter to take on a warm golden brown hue with a nutty aroma. Remove from heat immediately—transfer and cool. Transfer the brown butter to a heatproof bowl. Allow the brown butter to come to room temperature before using.
Make the Brown Butter Chocolate Chip Cookie Dough: Whisk to combine the flour, baking soda, and cornstarch in a large bowl; set aside. Combine the brown butter and sugars in a stand mixer fitted with the paddle attachment. Beat on low speed until well combined, about 2 minutes; the mixture will look grainy. Add the eggs one time, beating after each addition until incorporated. Add the both types of vanilla.
Scrape down the side of the bowl as needed. Reduce the speed to medium, add the flour mixture and beat until just incorporated. Finally, stir in the chocolate chips and nuts if using. Transfer the cookie dough into a medium bowl, cover it tightly, and chill in the refrigerator until firm, about 30 minutes to 1 hour. If chilling for 3+ hours, make sure you let the cookie dough sit at room temperature for at least 30 minutes before rolling into balls; the cookie dough will be very stiff after being in the fridge that long.
Form and bake the cookies: Preheat the oven to 350 °F. Position racks in the upper and lower thirds of the oven. Line two baking sheets with parchment paper; set aside. If you only have 1 baking sheet, let it cool completely between batches.
Using a 2-inch (2 tablespoons)cookie scooper, scoop the dough, scraping each against the bowl as you scoop. Roll each mound in your hands to form a ball.
The dough will be very soft, so handle carefully and work quickly. Drop each ball into the cinnamon and sugar mixture and roll around to coat thoroughly. Place on the prepared baking sheet, about 2 to 2 inches apart. Bake one sheet at a time until the cookies have puffed up and the tops begin to crackle, 10 minutes; do not overbake.
Remove from the oven and let cool slightly on the baking sheet, then transfer the cookies to a wire rack to cool completely. Repeat forming the remaining dough into balls. Store the walnut chocolate chip cookies in an airtight container.