In a 6-quart Dutch oven, heat 2 tablespoons EVOO over medium heat. Working in batches, add the pork, season with salt and pepper and brown on all sides, about 5 minutes per batch. Transfer to a bowl.
Pour off the fat from the pan. Add the onion, garlic, oregano, anchovies and remaining tablespoon EVOO to the pan and season with salt and pepper. Cover and cook over low heat, stirring occasionally, until softened, 8-10 minutes. Add 2 cups of water and bring to a boil over medium-high heat, stirring to scrape up any browned bits from the bottom of the pan. Add pork, fennel bulb and stems, and the apricots, season again and press a piece of parchment on top. Cover and cook over low heat until meat is very tender, about 1 1/2 hours. Discard the fennel stems, ladle the stew into large, shallow bowls and garnish with fennel fronds.