Preheat the oven to 350 degrees Fahrenheit (180 degrees Celsius)
Generously grease a bundt or babka pan, making sure to get into all the grooves.
In a medium bowl whisk together flour and baking powder. Set aside.
Crack 4 fresh eggs into the bowl of a standup mixer and add ¾ cup sugar. In a large bowl or stand mixer, beat the eggs and sugar on medium-high speed until pale, thick, and fluffy, about 7–10 minutes.
With the mixer going on low, add ¾ cup oil in a slow stream. Turn off the mixer.
Add 1 ¼ cups plain yougurt, lemon zest and vanilla, and fold it in gently with a silicone spatula.
Add the flour mixture in 4 additions, folding gently after each addition just until combined. Do not overmix.
Transfer the batter into the prepared bundt cake pan and smooth out the top.
Bake in the preheated oven at 350 degrees for 35-40 minutes or until a toothpick comes out clean. Don't overbake.
Place the pan on a cooling rack and let it rest for at least 10 minutes. Invert the pan carefully and the babka should come right out. Let cool completely before icing.
If needed, trim the bottom slightly with a serrated knife for a flat base.
To make the icing place lemon juice in a small to medium bowl. Add ½ cup of powdered sugar and mix with a spoon until smooth. Then add the remaining ½ cup of sugar and keep mixing breaking down any lumps with the back of the spoon until you reach the desired consistency. Drizzle evenly over babka.