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1
Make the pastry cream first so it has time to thicken fully. Heat the milk with the vanilla pod and seeds until just below the boil. Whisk the egg yolks with the sugar until pale, then whisk in the flour or starch. Pour the hot milk over the yolk mixture in a thin stream, whisking constantly, then return the whole to the pan over medium heat. Stir without stopping until the cream thickens, pulls away from the sides, and holds a firm shape on the back of the spoon — it must be dense enough to remain inside a warm, slightly soft shell without migrating. Press cling film directly onto the surface and set aside.
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2
Bring the water, butter, and salt to a full rolling boil in a heavy saucepan. Remove from the heat and add the sifted flour in one movement. Beat vigorously with a wooden spoon until the dough pulls cleanly from the sides of the pan and forms a smooth, cohesive mass. Return to low heat and continue beating for one minute to dry the dough further. Remove from the heat and allow to cool for five minutes.
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3
Incorporate the eggs one at a time, beating thoroughly after each addition. The dough is ready when it is smooth, glossy, and falls from the spoon in a slow, reluctant ribbon — neither stiff nor slack.
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4
Heat the oil in a deep, wide pan to exactly 170 °C and maintain it there throughout frying — this is where the dish lives or dies: a temperature above 170 °C sets the exterior crust before the interior has expanded fully, producing dense balls rather than hollow shells; oil that is too cool causes the bignè to absorb fat instead of puffing. Use a thermometer at all times. Working in small batches of five or six, spoon or pipe walnut-sized portions of dough directly into the oil, leaving ample room for each piece to rotate freely. The bignè will expand to approximately three times their original volume. Fry, turning occasionally, until deeply golden and hollow-sounding when tapped with a slotted spoon — about 6 to 8 minutes.
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5
Lift the bignè with a slotted spoon and drain briefly on kitchen paper. While still warm, roll each one in icing sugar — the sugar adheres to the lightly oily surface and forms the characteristic powdered exterior.
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6
While the bignè remain warm and the exterior yields slightly under pressure, pierce each one with a small nozzle and pipe in the thick pastry cream until the shell feels gently taut but does not bulge. Cold, rigid shells will crack during filling; over-filled shells will burst on the first bite. Serve within 30 minutes: beyond that point, moisture from the cream softens the fried exterior irreversibly.