Preheat oven to 350°
Toast Walnuts Dump walnuts on a parchment lined baking sheet. Toast in oven 8-10 minutes (if there are very small pieces, check after 5), until the nuts are very toasted and smell fragrant. Next, brown your butter. While ovens are toasting, brown your butter.
Brown Butter: Melt your butter in a small skillet on medium-low heat. Keep heating until it bubbles & then foams. Keep an eye on the butter, brown flecks will start popping up. Stir often. Remove from heat once lots of golden flecks are formed. Pour into a bowl to stop cooking process. Let butter come to room temp*
Mix Dry Ingredients: In a large bowl add flour, oats, & baking soda and whisk to combine. Mix Wet Ingredients: In a stand mixer with paddle attachment or large bowl with hand mixer, add room temperature brown butter & beat for 1-2 minutes on high. Add sugars & beat on high another 2 mins. On low, add eggs, egg white & vanilla extract. Beat on low or medium low until combined. Combine: Add dry ingredients to wet on low.
Add Chunks & Chill : Continue folding in any leftover flour with a rubber spatula. Add chocolate chunks & walnuts and fold in. Dough will be sticky. Chill dough for 30 minutes (you can leave it in the bowl).*
Bake: Make large balls (3ish tablespoons) and place them on parchment lined baking sheet. Very lightly press down the top. Bake 12 minutes until edges are golden & the center will not feel done if you tap the center. Let them cool for about 5 minutes on the baking sheet, and then place on wire rack to cool completely.