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1
Combine the flour, egg, warm water, sunflower oil, and salt in a bowl. Turn the mass onto a work surface and knead vigorously for 10 minutes until the dough is completely smooth, supple, and springs back when pressed. The dough must be worked long enough to develop the gluten chains that allow extreme stretching without tearing — this step cannot be shortened.
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2
Shape the dough into a ball, place it under a warm bowl, and leave it to rest for 1 hour at room temperature (22–24 °C). The gluten must relax fully; a cold room or shortened rest will cause the dough to resist stretching and tear.
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3
While the dough rests, combine the sliced apples in a bowl with the Armagnac, 100 g of the sugar, and the vanilla. Turn the apples to coat evenly and leave to macerate for 2 hours. The Armagnac defines the Gascon character of the dish; this step is not optional.
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4
Cover a large table with a clean cloth and dust it generously with flour. Lay the rested dough in the centre. Working with the backs of the hands — never the fingertips — ease the dough outward from the centre, moving around the table, until it stretches to near-transparency across the entire cloth and hangs over the table edges without tearing. It must be transparent enough to read newsprint through it. Trim and discard any thick border edges. This is where the dish lives or dies: insufficient gluten development or haste at this stage produces torn sheets that cannot be layered.
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5
Brush the stretched sheet generously with melted goose fat. Fold or cut the sheet to create 8–10 manageable layers. Butter a deep mould or baking sheet. Lay the first layers into the mould, brushing each with goose fat and dusting lightly with the remaining 50 g of sugar to encourage caramelisation between the layers. Distribute the macerated apple slices across the interior layers.
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6
Bring the uppermost sheets up and over the filling, gathering and ruffling them loosely to form an ornamental dome. Brush the dome thoroughly with the remaining goose fat. The ruffled surface will crisp and caramelise in the oven.
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7
Bake at 190 °C for 35–40 minutes. The pastis is ready when the dome is deep golden and the top layers are shatteringly crisp to a light tap. The kitchen will carry the scent of caramelised sugar, warm Armagnac, and rendered fat. Serve warm, within 2 hours of baking.