Make the flax egg by mixing 2 tbsp of ground flax seeds and 5 tbsp of water in a bowl, set aside.
Heat the oven to 220°C and line a 12-count muffin pan with cupcake liners.
To make the streusel topping combine 1 tbsp sugar, 1/3 cup brown sugar, 1 tsp cinnamon, and 1/4 cup of melted butter in a medium bowl. Mix well and then add 2/3 cup of all-purpose flour and mix together. The streusel topping will be thick and crumbly. Set aside for later.
In a bowl add 1/2 cup of brown sugar, 1/4 cup of sugar, and 1/2 cup of softened vegan butter. Beat with a handheld or stand mixer until smooth and creamy. Scrape down the sides and add the flax eggs, 1/2 cup of greek style yogurt, and 1 tsp of vanilla extract. Mix again until well combined.
Cut the apples into small cubes and set aside.
In a separate bowl sift 1 and 3/4 cup all-purpose flour, 1 tsp baking soda, 1 tsp baking powder, 1 tsp ground cinnamon, and 1/2 tsp salt together. Then add the dry ingredients to the wet mix. Then add the soy milk and chopped apples and mix with a spatula.
Spoon the batter evenly into the muffin pan, filling each almost all the way to the top. Then divide the streusel topping evenly between the muffins.
Bake for 5 minutes on 220°C then lower the heat to 180°C and bake for further 15-18 minutes or until you can stick a toothpick in the middle and it comes out clean.
Remove the muffins from the muffin tin and allow completely cool.
Make the vanilla icing: Whisk together all the ingredients and drizzle over cooled muffins.