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1
Place the sliced garlic and goose fat in a wide, heavy saucepan over the lowest possible heat. Sweat, stirring occasionally, until the cloves are soft, translucent, and pale gold — 12 to 15 minutes. The garlic must not colour; the moment browning begins, bitterness overtakes the sweetness the dish requires. The cue is a sweet, mellow aroma with no sharpness.
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2
Sprinkle in the flour and stir to coat every piece of garlic evenly. Cook the roux for one minute, still over gentle heat, until the raw-flour scent disappears and the mixture looks sandy.
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3
Add the water in a thin, steady stream, whisking to prevent lumps. Raise the heat to bring the soup to a bare simmer, then reduce and cook, uncovered, for 25 to 30 minutes. The garlic must dissolve entirely into the broth, leaving no discernible pieces; the surface will carry a pale, ivory sheen. This is where the dish lives or dies: insufficient cooking leaves raw garlic texture and harsh flavour; the full simmer produces the sweet, mellow, penetrating character that defines Tourin Gascon.
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4
Season with salt, white pepper, and a splash of vinegar. Taste: the acid should brighten the garlic's sweetness without announcing itself. Adjust as needed.
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5
Beat the egg yolks in a bowl. Ladle the hot broth into the yolks in a thin, continuous stream, whisking constantly (tempérer), until the yolk mixture is warm and fluid. Return the tempered liaison to the pot over very gentle heat and stir without stopping until the soup reaches 80°C and coats the back of a spoon with a silky, velvet sheen. Remove from the heat immediately: above 85°C the eggs scramble and the texture becomes grainy.
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6
Lay one slice of oven-dried stale bread in each warmed bowl. Ladle the soup over the bread — the bread should absorb the broth steadily without disintegrating. Finish with a drizzle of goose fat or walnut oil and serve at once.