Soak salt cod in cold water for about 12 hours, changing the water twice.

Divide peppercorns, bay leaf, and thyme into two batches, wrapping each in cheesecloth sachets.

Reserve one clove of garlic and slice the remaining head in half crosswise.

Pour cream into a small saucepan, add half the garlic head, bring to a boil, reduce heat, and infuse for 45 minutes.

Pour 1/4 cup olive oil into another small saucepan, add the other half of the garlic head, heat until it sizzles, reduce heat, and infuse for 45 minutes.

Strain the cream and oil separately, keeping them warm.

Combine 2 cups of milk, a herb sachet, and the cod in a saucepan. Bring to a boil, lower heat, and cook until the cod flakes easily, about 15 minutes.

While the cod simmers, cook potatoes with the remaining milk and herb sachet. Bring to a boil, and simmer for 10 to 15 minutes, until the potatoes are tender.

Drain the cod and potatoes, discard the sachets, and transfer to a food processor.

With the machine running, slowly add the infused oil, followed by the cream.

Pulse until smooth, then season with white pepper to taste.

Use a small spoon or piping bag to stuff 8 piquillo peppers with the brandade mixture.

Cover with plastic wrap and refrigerate until ready to serve.

Cut the remaining piquillos into thin strips.

Crush and peel the reserved garlic clove.

Warm 1 tablespoon of olive oil in a skillet over moderate heat.

Add the sliced piquillos, red pepper, shallot, and garlic to the pan, season with salt and pepper, and sauté for 2-3 minutes, stirring.

Add 2 tablespoons of water, cover the pan, and lower the heat.

Cook for about 20 minutes, or until the vegetables are completely tender.

Discard the garlic and transfer the mixture to a blender. Puree, then strain through a sieve and keep warm.

Preheat the oven to 250 degrees.

Warm 1 tablespoon of olive oil in a skillet over medium heat.

Add the stuffed peppers, and transfer to the oven for 20 minutes.

Ladle pepper puree onto plates and top with the baked stuffed peppers to serve.