Preheat your oven to 350°F (175°C) and grease a 9x13 inch baking pan or line it with parchment paper for easy removal later.
In a medium saucepan, melt the butter over low heat. Once melted, remove from heat and stir in sugar until well combined.
Allow the mixture to cool slightly, then add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
In a separate bowl, sift together the flour, cocoa powder, and salt. Gradually fold this dry mixture into the wet ingredients using a spatula, being careful not to overmix.
Once just combined, fold in the semi-sweet chocolate chips and walnuts (if using) until evenly distributed throughout the batter.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake in the preheated oven for 25-30 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs (not fully clean for that fudgy texture!).
Remove from the oven and let cool in the pan for about 10 minutes before transferring to a wire rack to cool completely.
Once cooled, cut into squares and enjoy your delicious fudgy brownies!