Preheat oven to 200 °C / 400 °F. Line two baking sheets with parchment; dab trays with butter so parchment sticks.
In a medium saucepan combine water, milk, and butte. Bring to a gentle boil; butter must fully melt.
Off heat, dump in the flour, baking powder, sugar, salt, and vanilla. Stir vigorously until a dough ball forms.
Return pan to medium heat. Cook 4–5 min, stirring constantly, until dough leaves a thin film on the bottom. Transfer to stand‑mixer bowl; cool 5 min.
Beat dough on low; add 4 eggs one by one, waiting for full absorption. Dough should be glossy, thick yet pipeable—forms a V‑shaped ribbon off the spatula.
Scoop dough into a large piping bag. Pipe walnut‑size mounds on trays. Generously shower with pearl sugar.
Bake 15–20 min until puffed and deep golden. DO NOT open oven early or they’ll collapse (wait about 5 to 10 minutes).
They taste best on the day they are made.