Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Melt the chocolate chips, butter, and peanut butter together in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Stir vanilla into chocolate mixture.
Stir flour, sugar, eggs, baking soda, and salt into the chocolate mixture. Fold walnuts into batter. Pour batter into prepared baking dish.
Bake brownies in the preheated oven until a toothpick inserted in the center comes out with moist chunks on it, but not wet, about 30 minutes. Cool to room temperature before cutting into squares.