Spread 2 tablespoons of salt on a large plate. Set the cod on the plate and pat 2 tablespoons of salt on top. Cover with another large plate and weigh it down with a heavy can. Let stand at room temperature for 30 minutes.
Rinse the cured cod. In a medium skillet, combine the cod with the bay leaf, peppercorns and 4 cups of water. Bring to a simmer, then cook over low heat for 5 minutes. Using a slotted spoon, transfer the cod to a bowl; let cool slightly and shred it. Strain the cooking liquid through a sieve into a glass measuring cup; discard the contents of the sieve. (The cooked cured cod and cooking liquid can be refrigerated, separately, overnight.)
In a medium saucepan, warm the oil over moderate heat. Add the onion and garlic and cook, stirring occasionally, until softened, about 5 minutes. Add two-thirds of the chickpeas and 3 cups of the cod cooking liquid and bring to a simmer; reserve the remaining cod cooking liquid. Reduce the heat to low and simmer the soup, partially covered, until the chickpeas are very soft, about 30 minutes.
Meanwhile, poach the eggs: Bring a medium skillet of water to a simmer. Crack 1 egg at a time into a small bowl; then gently slide each egg into the water. Cook over moderately low heat until the whites are set and the yolks are still soft, about 3 minutes. Using a slotted spoon, transfer the eggs as they’re done to a paper towel–lined plate to drain.
In a blender, puree the chickpea mixture until very smooth. The soup should be thick but pourable; add a little of the reserved cod cooking liquid if necessary. (At this point, the soup can be refrigerated overnight.) Return the chickpea soup to the pan and reheat gently. Add the spinach and remaining whole chickpeas and cook until the spinach wilts, about 1 minute. Taste for seasoning, keeping in mind that the cured cod is salty.
Divide the shredded cod among soup bowls. Ladle the soup on top and add a poached egg to each bowl. Garnish with parsley, drizzle with oil and serve.