Lightly grease an 8-inch by 8-inch by 2-inch baking pan.
Place the butter and chocolate in a double boiler or microwave-safe bowl and melt over low heat, stirring occasionally. Once the butter is melted, take off heat to stir the chocolate to help it melt. Don’t let the mixture get too hot or the chocolate will seize.
Stir in the sugar and vanilla extract, which helps cool the chocolate.
Add eggs and vanilla and beat lightly. Beating too hard will incorporate air and create a less dense brownie.
Stir in the flour mixing until all the flour has been incorporated.
Stir in the chips and nuts, if using, until thoroughly blended.
Spread mixture into the greased baking pan.
Top the brownie batter with the walnuts, if desired.
Bake in preheated oven for 25 minutes; the brownies are done when top looks cracked and the sides have pulled away slightly. A toothpick inserted in the middle should show a moist crumb.