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1
In a wide, heavy pan set over medium-low heat, warm the olive oil and duck fat together. Add the pepper strips, sliced onion, and minced garlic. Cook slowly for 25–30 minutes, stirring occasionally. The peppers are ready when they are completely soft and beginning to melt into one another — no trace of crispness should remain.
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2
Add the peeled, seeded, and chopped tomatoes and the piment d'Espelette. Continue cooking over medium-low heat for a further 15–20 minutes. This is where the dish lives or dies: the tomatoes must dissolve entirely into the peppers, losing all individual identity, and most moisture must evaporate, leaving a thick, jammy fondue. Season with salt.
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3
Pour the lightly beaten eggs over the hot fondue. Stir gently and continuously over medium-low heat in the manner of oeufs brouillés until the eggs form soft, creamy curds bound within the pepper mixture — not cooking separately from it but becoming one with it. Remove from heat while the eggs remain slightly underset; residual heat will complete the cooking.
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4
Transfer immediately to warmed plates or serve directly from the pan. Drape thin slices of jambon de Bayonne over the surface and accompany with thick slices of country bread. Piperade does not hold or reheat; it must be served at the moment it is made.