Add butter, granulated sugar and brown sugar into a large bowl. Mix with a hand mixer on low until smooth.
Add 2 tablespoons of egg. Keep mixing until fully and combined.
Add flour, baking powder, baking soda, and salt. Mix until it just forms a dough. Do not over mix. It is OK if there is a bit of dry flour left in the bowl.
Change to a spatula. Add walnuts and add them into the dough.
Preheat the oven to 180 degrees C (350 F).
Line a cookie sheet with parchment paper or prepare a nonstick cookie sheet. Scoop about 1 and 1/2 tablespoons cookie dough and transfer it to the cookie sheet, 1 scoop apart from each other. You can also use smaller cookies by using 1 tablespoon dough per cookie.
Once the dough is divided, flatten the dough into a round disk, about 6-cm (1/3-inch) in thickness.
Mix the remaining egg with 1 teaspoon water to make the egg wash. Right before baking, brush a thin layer of egg wash on the top of each cookies.
Bake for 31 to 35 minutes, or until the cookies are cooked through (*see footnote 2).
Remove the cookies from the oven and place them onto a cooling rack for 5 minutes. Then transfer the cookies onto the rack until completed cooled down.
You can store them in an airtight container for up to 3 days.