Position one of your oven racks in the middle of the oven.
Preheat the oven to 350F degrees. Grease a 9-inch square baking pan with nonstick cooking spray or shortening.
In a small saucepan, melt the butter and chocolate over low heat, stirring frequently. Remove from the heat and cool slightly, about 5 minutes.
In a large bowl beat the sugar, vanilla, and egg with an electric mixer on high speed for 5 minutes, scraping down the sides of the bowl with a rubber spatula if necessary to make sure everything is well blended.
Beat in the chocolate mixture with the mixer on low speed, scraping the bowl occasionally as necessary.
Add the flour and beat on low speed just until blended.
Spread batter into the prepared pan.
Bake 40 to 45 minutes or just until the brownies begin to pull away from the sides of the pan and center is set.
Remove from the oven and place the pan on a wire rack to cool completely, about 2 hours.