Preheat the oven to 325 only - not 350. If you are using glass, then preheat to 300 only.
Put enough foil in an 8x8 pan to go over the sides. Grease and flour the foil. Trust me, you'll be happy you did.
Put the chocolate and butter into a double boiler. Melt, undisturbed until the chocolate has melted. Then you can stir them together. Remove from the heat.
Mix together the sugar, egg, and vanilla.
Slowly add the chocolate-butter mixture, beating it until all is incorporated.
Finally, add the flour carefully and beat it until it, too, is fully incorporated.
Pour into the 8x8 prepared pan.
Put in the oven and bake for 40 minutes. Check at 30, but the brownies will be done at 40 minutes.
Remove using the sides of foil and place on a rack to cool.
Peel off the foil and cut into squares as desired.
Notes
You can use a microwave, but I trust this method a little more. It's a gentler, safer way to melt chocolate without the possibility of burning. My son put a little Nutella on one of his and said it was delicious. If you want to frost these, have at it, but they really don't need it.
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