In a small saucepan, melt the vegan butter over medium heat, then add the plant milk.
Mix in the sugar, then check the temperature (it should be warm, but not hot: i.e. dip your pinky finger in it, it should not be uncomfortable).
Sprinkle in the yeast, and let it sit for 5 minutes until frothy.
Add the flour to a medium-sized bowl, then add the yeast mixture, aquafaba, vanilla, and salt.
Mix until combined, then knead for 5 minutes by hand, or 2-3 minutes with a dough mixer (I used a food processor with a dough blade).
Place in a bowl and cover with a kitchen towel, and leave to rise for one hour (it should double in size).
Tip out onto a flat surface that's lightly dusted with flour, and cut into 8 equal parts (like puffy pizza slices).
In the palm of your hands, roll each piece into a ball.
Space the dough balls apart, cover with the kitchen towel, and leave to rise for 2-3 hours at room temperature.
Pour enough oil into the pan so it's about 5 cm deep, and heat on medium heat.
The oil should be ready at 190ºC, but if you don't have a thermometer (like me), test it with a small piece of dough: if it sizzles and goes golden brown after 2 minutes, it's a good temperature. If the oil is not hot enough, the dough will soak it up and get all greasy; if it's too hot, the dough will burn.
Carefully drop each dough ball into the oil with a slotted spoon or a fork, giving them enough space to float around.
Fry for 2-3 minutes, until the underside is golden, then flip and fry for another 2-3 minutes. Don't move them around while frying.
Transfer the fried bollen (yes, I can't bring myself to say 'balls' again) to plate lined with paper towels.
Let them cool for a few minutes.
Meanwhile, pour the sugar into a bowl, and fill your piping back (or big syringe, like me) with jam.
When the Berliner are cool enough to touch, poke the end of a wooden spoon or chopstick into the ball and move it around a bit to make a cavity.
Stick your piping tip into the hole, and fill the Berliner with some jam (you be the judge).
Once they're all filled, take them one-by-one and roll them around in the sugar.
Place them on a plate, and dust with powdered sugar to serve!