Preheat the oven to 350 degrees F. Grease a 9-inch square baking dish with 1 tablespoon of the butter and set aside.
In a saucepan, melt the remaining 8 tablespoons butter and the chocolate over medium-low heat, stirring occasionally with a wooden spoon.
In a large bowl, combine the sugar and hot chocolate mixture, whisking until smooth. Add 1/4 cup of the flour and whisk to combine. Add 1 egg and whisk to combine. Add the remaining flour and egg and whisk to combine. Stir in the vanilla extract. Pour into the prepared dish and tap the bottom to release any air bubbles. Bake until set and a toothpick inserted into the center comes out clean, 25 to 30 minutes.
Let cool completely before cutting into 9 equal pieces.
Yield: 1 (9-inch) pan, 6 to 9 servings