Preheat oven to 350 degrees. Brush a 9-inch square baking pan with oil; line with a strip of parchment paper, leaving an overhang on two sides, and brush paper with oil.
In a medium bowl, whisk together flour, cocoa, salt, and baking soda; set aside. In another medium bowl, whisk together brown sugar, applesauce, sour cream, melted chocolate, eggs, and oil until combined; add flour mixture, and mix just until moistened.
Spread batter in prepared pan; sprinkle top with chopped chocolate. Bake until a toothpick inserted in center of cake comes out with a few moist crumbs attached, 30 to 35 minutes.
Cool completely in pan. Use paper overhang to lift cake from pan; peel off paper, and discard. Cut into 16 squares.