In a large Dutch oven or casserole with tight-fitting lid, heat 1/4 cup of the olive oil over medium-high heat until very hot. Working in batches, cook the pork until golden brown on at least two sides, about 10 minutes per batch, reducing the heat if it threatens to burn. Transfer to a bowl. Toss the pork generously with salt and set aside.
Let the Dutch oven cool slightly and place over medium heat. Add half of the onion and half of the minced garlic. Cook, stirring occasionally, until pale gold at the edges, about 8 minutes. Add the wine, scraping the bottom of the pot to loosen any browned bits. Simmer gently until reduced by about one-third, about 5 minutes. In a bowl, combine the tomato paste and water, stir to dissolve, and add to the pot. Stir in the bay leaves, thyme, paprika, 1/2 teaspoon salt, and the pork and any accumulated juices. Bring just to a simmer, cover the pot, and cook until the pork is tender, about 1 1/2 hours, adjusting the heat as needed to maintain a gentle simmer.
Rub the bread slices with the garlic halves and drizzle with a bit of olive oil. Toast or broil the bread until crisp but still tender inside and set aside.
Add the mustard greens to the pork stew a handful at a time, stirring until each handful wilts. Stir in the sweet pepper paste and a few generous grindings of black pepper. Cover and keep warm over low heat.
In a medium saucepan with a tight-fitting lid, heat the remaining 1/4 cup olive oil over medium heat. Cook the remaining onion and minced garlic, uncovered, until pale gold at the edges, about 8 to 10 minutes. Add the cockles, raise the heat to high, cover, and steam, shaking the pot occasionally, until they open, 3 to 4 minutes. (If you're using littleneck clams, they'll take a little longer.) Discard any that don't open.
Transfer the entire contents of the saucepan to the pork stew, stirring gently to combine without knocking the cockles from their shells. Cook gently for 1 minute, adding salt if necessary. Divide the stew and broth among 4 warmed shallow soup bowls, distributing the cockles attractively around the bowl. Drizzle each serving with additional olive oil and sprinkle with the cilantro. Garnish with the garlic toasts and serve immediately.