Melt butter over low heat. Pour into a medium bowl and cool a few minutes.
Beat eggs separately in a small bowl.
Stir sugar into melted butter, followed by flour, cocoa powder, vanilla, and eggs. Mix just to combine.
Pour into buttered 8x8-inch pan and bake in preheated oven for 20-25 minutes. For a fudgier brownie, keep the baking time at the lower end of the range. For a cakier consistency, add 1 teaspoon of aluminum-free baking powder to batter along with flour.