For the dough: Combine water, milk and vanilla in a saucepan and bring to a boil. Combine poppy seeds and flour, then stir into hot liquid until a large lump forms.
Put dough into a bowl, then stir in 1 egg. Let dough cool slightly, then stir in remaining eggs.
Pour dough into a piping bag with a large star tip. Pipe rings of dough onto a baking sheet lined with parchment paper.
Heat frying oil to 175°C (approximately 350°F). Gently slide rings into hot oil and cook until golden brown on both sides. Remove, drain on paper towels and let cool.
Whisk together powdered sugar and lemon juice to form an icing. Coat doughnuts with icing. Let set, as serve warm or at room temperature.