Lightly butter a sheet of parchment paper, then use the parchment paper to line an 8x8-inch baking pan (the parchment should go up the sides of the pan, too). Set aside.
In a saucepan over medium-low heat, melt the butter and chocolate, stirring frequently. Remove from heat when chocolate is just melted. Add in the sugar and vanilla extract and stir to combine. Beat in the eggs, one at a time. Beat in the flour and stir vigorously by hand for 1 full minute. During this time the texture of the batter will go from grainy to smooth and glossy. The batter should also start to pull away from the sides of the pan.
Transfer the batter to the prepared pan and spread evenly. Bake for 30 minutes. Do not overbake. Remove from oven and cool completely in the pan before cutting into squares.