Preheat the oven to 350° and lightly oil a 12 cavity doughnut pan (or a couple of 6 cavity pans).
To make the crumble topping, stir the flour, brown sugar, cinnamon, and salt together in a small bowl. Stir in the melted vegan butter until fully mixed. Place the bowl in the refrigerator and allow the mixture to chill while you prepare the doughnuts.
To make the doughnuts, first whisk the flour, brown sugar, baking powder, baking soda, cinnamon, ginger, nutmeg, cloves, and salt together in a large mixing bowl.
In a separate bowl or large liquid measuring cup, stir together the applesauce, milk, oil, and vanilla.
Add the applesauce mixture to the dry ingredients and stir, just until fully mixed.
Fold in the diced apple.
Spoon the batter into the doughnut pan cavities, distributing it evenly and smoothing out the tops of the doughnuts. Sprinkle the topping over the batter, gently pressing it in with your hands.
Bake the doughnuts for about 12 minutes, until the tops spring back when lightly touched. Transfer the pan(s) to a cooling rack and allow the doughnuts to cool completely.
When the doughnuts are cool, stir together the frosting ingredients in a small bowl. Add more powdered sugar if the mixture is too thin, more milk if it's too thick. You want to just be able to drizzle it.
Remove the doughnuts from the pan(s), gently loosening them with a butter knife, if needed. Place the doughnuts crumb side up on a dish or work surface and drizzle them with the glaze.
Serve.