Preheat the oven to 160C (320F) and line a muffin pan with 9 paper liners.
In a mixing bowl, combine the flour baking powder, salt, and poppy seeds.
In another mixing bowl, whisk together the eggs, granulated sugar, milk, lemon juice, zest, butter, oil, and vanilla extract.
Add wet ingredients to dry ones and mix just until no dry ingredients are left.
Divide the batter between muffin liners and bake for 25-27 minutes or until a toothpick inserted into a muffin comes out clean.
Remove muffins onto a cooling rack to cool them down to room temperature.
In a bowl, whisk together the sugar icing ingredients and drizzle on top of completely cooled muffins. Optionally, sprinkle lemon zest on top so it sticks, and serve.