Preheat the oven to 325°F (165°C) with a rack in the center. Line the bottom and sides of an 8-inch square baking pan with parchment paper or aluminum foil, allowing an overhang on two opposite sides for easy removal.

Place the butter in a medium saucepan over low heat. Cook gently until fully melted, then remove from the heat immediately. Do not let the butter brown.

Stir the granulated sugar, cocoa powder, vanilla extract, and salt into the hot butter. Mix well until everything is combined. The mixture will look grainy at this stage, but it will smooth out once eggs are added.

Set the pan aside and let the mixture cool until it is warm to the touch, about 5 to 10 minutes. You want it warm, not hot, so it does not scramble the eggs.

Add the cold eggs one at a time, beating vigorously after each addition until the batter is thick and glossy.

Sprinkle the flour over the batter and fold it in with a wooden spoon or spatula. Beat for 40 to 50 strokes until the batter is very thick and pulls away from the sides of the pan.

If using nuts or chocolate chips, gently fold them in at this point.

Scrape the batter into the prepared pan and spread it evenly into the corners. Bake for 20 to 30 minutes, until the edges are set and the center is still slightly underdone. A toothpick inserted into the center should come out with a few moist crumbs.

Let the brownies cool completely in the pan on a wire rack. For clean slices, use a sharp chef's knife and wipe the blade between cuts. You can also chill the brownies in the refrigerator for 1 to 2 hours before slicing.