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1
Heat the oven to 200 °C. Roll each portion of pâte brisée into a disc 3 mm thick — no thicker. Lay the first disc flat on a baking sheet lined with baking paper. The disc should be uniform and without cracks; press any splits together before proceeding.
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2
Peel, core, and slice the apples to 2–3 mm. In a bowl combine the apple slices with the sugar, flour, and the scraped seeds of the vanilla pod (or rum). Toss until every slice is evenly coated. The flour must be present; without it the base will be sodden.
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3
Spread the apple mixture evenly over the base disc, leaving a 2 cm border clear at the edge. Dot the cold butter pieces across the surface. Lay the second pastry disc over the apples and press the edges together firmly with a fork to seal. The pompe must sit no taller than 4 cm — it is a flat galette, not a raised pie. Cut several small steam vents in the top. The seal and the thinness of the pastry decide whether the dish is crisp or collapsed: this is where the pompe lives or dies.
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4
Brush the top disc lightly with crème fraîche. Transfer to the oven and bake for 25–30 minutes, then brush again generously with crème fraîche and return to the oven for a final 5 minutes. The top is ready when it is deeply golden and the cream wash has dried to a matte, toasty finish — not shiny, not pale.
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5
Remove from the oven and rest for 5 minutes. The pompe should feel firm but not rigid when pressed lightly at the centre, indicating the apples have cooked through. Cut into wedges and serve warm.