In medium saucepan, melt 1/3 cup butter over medium heat. Once melted, cook for about 2 minutes until butter just starts to turn brown. Remove from heat. Immediately add sugar, applesauce and vanilla, combining well.
While butter is melting, in large mixing bowl, combine flour, baking soda and cinnamon.
Add warm butter mixture to the dry ingredients. Combine well and pour batter into prepared cake pan.
Cook 30-35 minutes until toothpick inserted comes out clean.
Prepare glaze: Melt butter (in microwave or stovetop). Add remaining ingredients, combining well until smooth. Let cool about 5 minutes to allow some thickening; glaze will be slightly thin. Pour over warm cake.
Let set about 30 minutes before serving. For nice garnish, lightly sprinkle ground cinnamon over frosted cake.