1. In a large bowl, beat together the butter and granulated sugar with an electric mixer on high speed 1 to 2 minutes, or until fluffy. Beat in the applesauce, cinnamon, vanilla, and eggs until well mixed. Beat in the melted chocolate until blended. Add the flour, baking soda, baking powder, and salt. With the mixer on low speed, beat in the dry ingredients, adding the buttermilk as you beat. Beat just until evenly mixed. By hand, stir in the chocolate chips and walnuts. Scrape the batter into a buttered 3 1/2-quart electric slow cooker and smooth the top.
2. Cover and cook on the high heat setting 2 1/4 to 2 1/2 hours, or until a cake tester inserted in the center comes out clean. (Do not cook on the low heat setting for a longer time.)
3. Remove the lid and let the cake stand in the slow cooker until just barely warm. To unmold, run a sharp knife around the inside edges of the cooker and with a large spatula, carefully lift out the cake in one piece. Sprinkle powdered sugar over the top.