Preheat oven to 350 °F and grease and flour a 9-x-5-inch loaf pan.
For loaf, whisk vegetable oil, brown sugar, sugar, whole egg, egg yolk and vanilla until smooth. Stir in applesauce. In a separate bowl, sift flour, baking powder, baking soda, nutmeg and ginger and stir gently into applesauce mixture.
For streusel, combine brown sugar, flour and cinnamon. Stir in melted butter and walnut pieces and toss to coat.
Spoon half of the applesauce batter into prepared loaf pan and sprinkle with half of the streusel. Top with remaining batter and sprinkle remaining streusel to finish. Bake for 40 to 50 minutes, until a tester inserted in the center of the cake comes out clean. Cool loaf for 20 minutes, before turning out on a rack to cool completely.