Preheat the oven to 325 degrees F. Grease a 9-by-9-inch baking dish with the softened butter and set aside.
In a medium saucepan over medium-low heat, melt the 2 sticks butter and the chocolate together, stirring constantly, until smooth. Remove from the heat and stir in the cocoa powder, granulated sugar, salt, 1 cup of the flour and 5 of the eggs. Pour the batter into the prepared baking dish.
Using an electric mixer, beat the cream cheese until smooth, 3 to 4 minutes. Add the confectioners' sugar, vanilla, the remaining 1/4 cup flour and the remaining egg. Mix until smooth. Dollop the cream cheese mixture over the brownie batter. With a butter knife or fork, swirl the mixtures together to form a design.
Bake until the brownies just slightly pull away from the sides of the baking dish, 45 to 50 minutes. Allow to cool in the pan for 15 minutes. Slice into 9 squares and serve.
You can make your own confectioners' sugar by streaming granulated sugar into the small opening of a blender set to high speed. One cup granulated sugar yields 3/4 cup confectioners' sugar.
Adapted from "Trisha's Table: My Feel-Good Favorites for a Balanced Life" by Trisha Yearwood © Clarkson Potter 2015. Provided courtesy of Trisha Yearwood. All rights reserved.