In a medium pot, bring 1 cup of water to a boil. Add the salt, sugar, allspice berries, peppercorns, and 1 bay leaf; stir to dissolve the salt and sugar. Remove from the heat and add 3 cups of cold water. Refrigerate until cold.
Lay the cod trimmimgs in a large roasting pan. Pour the chilled brine over the fish, completely submerging it in the brine. Cover and refrigerate for 2 hours. Remove, then discard the brine.
In a medium pot, add the cod and fish stock, completely submerging the fish. Set over low heat, and as soon as the stock begins to simmer, remove from the heat. Cover and let rest until the fish is cooked through, about 5 minutes. Use a slotted spoon to gently transfer the cooked cod to a plate. Carefully remove any skin and bones and discard.
Set a fine strainer over a large measuring cup and strain the stock; reserve.
In a medium pot over medium heat, melt the butter. Add the fennel and pimentón, and cook, stirring occasionally, until the fennel is slightly softened, 5–7 minutes. Add the onions, remaining bay leaf, and chile flakes, and cook 1 minute more. Add the reserved stock, bring to a simmer, and cook until the onions and fennel are tender, 12–15 minutes. Add the potatoes, and continue cooking until they are tender, 10–12 minutes. Add the half-and-half, and bring the mixture back up to a gentle simmer. Remove from the heat, and season to taste with kosher salt. Using your hands, flake the fish into large pieces and return them to the pot to reheat.
Serve immediately, or refrigerate up to 24 hours before reheating and serving.