If you didn’t buy pre-soaked salt cod fillets, soak them in cold water for 48 hours, changing the water every 8 hours. Once soaked, pat the fillets dry with paper towels.
In a deep, heavy-bottomed skillet, heat 2 cups extra-virgin olive oil over very low heat (about 140–160°F). Gently poach the fillets skin-side up for 8–10 minutes, until they release their gelatin. Transfer them to a plate and set aside, leaving the oil in the pan.
Raise the heat to medium-low and add the sliced garlic and dried cayenne chilies. Cook, stirring often, until the garlic is golden and fragrant, about 2–3 minutes. Remove and reserve the garlic and chilies.
Let the oil cool for about 5 minutes. Remove the pan from the heat (or reduce to the lowest setting), then slowly drizzle in the gelatinous juices released by the cod. Using a fine-mesh strainer or a whisk, stir continuously in a circular motion until the sauce emulsifies and becomes creamy. If the sauce breaks, whisk in 1 teaspoon of cold water until it comes back together.
Return the cod fillets to the pan, skin-side down, and heat for 2 minutes, gently swirling the pan to coat them with the sauce.
Serve immediately, garnished with the reserved garlic and chilies, and with crusty bread on the side for dipping.