Brownies: Make sure the oven rack is in the center position and preheat the oven to 350 degrees F. Grease a 9 by 13-inch baking pan with 1 tablespoon of the butter. Sprinkle 1 tablespoon of the flour into the pan and shake and swirl it to coat the sides and bottom evenly. Hold the pan over the sink or a trash can, then turn it upside down and lightly tap the bottom to release any extra flour. Set aside.
Place the remaining 1 1/2 sticks of butter and the chocolate in a large glass measuring cup or medium glass bowl, cover with plastic wrap, and microwave on high power for 1 minute. Remove from the microwave, uncover, and stir well. Return the butter and chocolate mixture to the microwave and cook 30 to 60 seconds, or until the chocolate is melted. (If you don't have a microwave, melt the butter and chocolate in the top of a double boiler.) Pour the melted chocolate mixture into a large mixing bowl. Add the granulated sugar and whisk until smooth. Add 1/4 cup of the flour and whisk to combine. Crack and add 1 egg and whisk to combine. Alternate adding the remaining flour and eggs, whisking well after each addition. Stir in the vanilla and toasted nuts and mix well. Pour the batter into the prepared pan. Holding the pan with both hands, give the bottom of the pan a quick rap against the countertop. Bake until the brownies rise and a toothpick inserted into the center of the pan comes out clean, about 30 minutes.
Using oven mitts or pot holders, remove the pan from the oven. Use the handle of a wooden spoon to poke holes into the brownies every 1 to 2 inches. While it is still warm, pour the Fudgy Topping over the brownies and smooth it with a rubber spatula. Let pan cool on a wire rack for at least 1 hour before serving brownies.
Yield: 24 (2-inch-square) brownies