Preheat the oven to 325 degrees F (165 degrees C). Butter and flour an 8x5-inch loaf pan.

Sift 1 cup plus 3 tablespoons all-purpose flour, 1 cup plus 3 tablespoons whole wheat flour, cornstarch, and baking soda together into a bowl.

Mix soy milk and vinegar together in a small bowl. Let stand until curdled.

Beat coconut oil in a stand mixer fitted with the paddle attachment until creamy. Add sugar and beat on medium speed until light and fluffy. Reduce speed to medium-low and beat in eggs one at a time. Scrape down the sides and bottom of the bowl. Stir in soy milk and vanilla on low speed. Add the flour mixture 1/2 cup at a time, mixing on low speed until batter is smooth.

Pour batter into the prepared loaf pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 60 to 70 minutes. Place pan on a wire rack on its side and cool for about 10 minutes.

Combine 1 cup confectioners' sugar, 2 tablespoons milk, and lemon juice in a small bowl to make glaze. Add more sugar or milk to reach the desired consistency.

Remove cake from the loaf pan. Drizzle glaze on top. Sprinkle coconut flakes evenly over the top.