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1
Combine the flour, salt, and sugar in a large bowl. Dissolve the yeast in the lukewarm milk and pour into the flour. Add the eggs and softened butter. Work the dough until it is smooth and comes away cleanly from the sides of the bowl. Cover and leave to rise at room temperature until doubled in volume.
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2
Heat a flat, shallow-patterned French gaufrier (not a deep Belgian iron). Divide the risen dough into small portions and cook each one, pressing the iron firmly closed, until the waffle is thin, pliable, and pale gold — not crisp. Remove each disc while still warm and pliable.
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3
Place the vergeoise, butter, rum, and scraped vanilla seeds and pod in a heavy saucepan over medium heat. Stir until the sugar dissolves, then continue cooking without stirring until the mixture reaches 110 °C (soft-ball stage) and forms a thick, toffee-like paste. This is where the dish lives or dies: below this temperature the filling will be too loose and will seep; above it, the filling will set hard and lose its characteristic chewiness.
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4
Working while both waffles and filling are hot, spread the toffee paste generously over one warm waffle disc. Lay a second disc on top and press firmly together. Trim the edges to neaten. The sealed sandwich should feel slightly resistant under the palm — soft and yielding, not stiff.
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5
Serve warm, at which point the vergeoise centre is soft and toffee-like. At room temperature the filling firms to a pleasant, dense chewiness — the texture that identifies an authentic gaufre fourrée lilloise.