Prepare an 8×8 baking pan by lightly spraying with cooking spray or greasing with butter. Line with parchment paper.
In a medium saucepan, melt butter over medium heat and cook until the butter is golden brown. Skim any foam from the sauce and discard. Whisk in the sugars. Remove from heat and whisk in the eggs, vanilla and pumpkin.
In a separate small bowl, whisk together the flour, cinnamon, baking powder, baking soda, salt and nutmeg. Stir into the butter mixture until just combined. Stir in half of the chocolate chips. Pour into prepared pan and spread evenly. Top with remaining chocolate chips and walnuts.
Bake for about 20 minutes or until it begins to pull from the edges of the pan. Remove from oven, cool completely (or at least until it is just lukewarm). Slice into 16 squares and serve.