Preheat the oven to 350 degrees F. Grease a 9- by 13-inch baking dish with 1 tablespoon of the butter.
For the batter: Heat a double boiler over medium heat. Add the butter and chocolate and stir until the chocolate has melted. Take off the heat and let cool slightly. Whisk in the brown sugar and eggs. Fold in the flour, walnuts and salt with a wooden spoon.
For the goat cheese mixture: Blend together the goat cheese, sugar and egg in a large bowl using an electric beater on low speed.
Pour the brownie mixture into the baking dish, and then evenly disperse spoonfuls of the cheese mixture over the batter. With a knife, score the surface into a pattern. Bake until a test toothpick comes out clean, 40 minutes.
Cool the brownies on a rack, and then cut into bars and refrigerate.