Preheat the oven to 350 degrees F. Grease and flour a 13x18-inch rimmed half sheet pan (or grease and line with parchment paper).
Place the butter, semisweet chocolate and unsweetened chocolate in a medium bowl and microwave on 50% power in 30-second increments, stirring after each, until completely melted and smooth. (Alternately, you can also melt the butter and chocolates in a double boiler on the stovetop). Allow to cool slightly.
In a large bowl, stir together the sugar, eggs, and vanilla with a wooden spoon. Stir the chocolate mixture into the egg mixture and allow to cool to room temperature.
In a medium bowl, sift together 1 cup of flour, the baking powder, and salt. Add to the cooled chocolate mixture and stir gently with the wooden spoon until mostly combined. In a medium bowl, toss the walnuts and chocolate chips with the remaining ¼ cup of flour, then add them to the chocolate batter and stir until totally incorporated. Pour into the baking sheet and spread into an even layer.
Bake for 20 minutes, then rap the baking sheet against the oven shelf to force the air to escape from between the pan and the brownie dough. Bake for an additional 15 minutes, or until a toothpick comes out clean. Allow to cool thoroughly, then refrigerate for at least 1 hour, and cut into 20 large squares. Leftover brownies can be stored in an airtight container in the refrigerator for up to 1 week.