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Place the pork shoulder in a large Dutch oven with the halved onion, garlic cloves, bay leaves, and 1 tsp salt. Pour in the chicken broth and water — the liquid should come at least halfway up the pork. Bring to a boil over high heat, skimming any foam that rises to the surface, then reduce the heat to a gentle simmer. Cover and cook for 1 hour 30 minutes, until the pork is fork-tender and an instant-read thermometer inserted into the thickest part reads at least 195°F (90°C).
While the pork simmers, char the poblano chiles directly over a gas flame or under a broiler, turning occasionally, until the skin is blackened and blistered on all sides, about 8–10 minutes. Transfer to a bowl, cover tightly with plastic wrap, and let steam for 10 minutes. Peel off the charred skin, remove the stems and seeds, and roughly chop the flesh.
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