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Preheat oven to 350 degrees. Season pork with salt and pepper. Heat a medium Dutch oven (6 to 8 quarts) over medium-high. Swirl in 2 tablespoons oil. Working in batches, brown pork all over, 16 to 20 minutes total (adding remaining oil when needed). Let pork cool slightly, then cut into 1-inch chunks.
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Return Dutch oven to medium-high heat. Add onion, season with salt and pepper, and sauté until translucent, about 3 minutes. Return pork to Dutch oven along with canned chiles, salsa, and 1/2 cup water. Bring to a boil, then cover and transfer to oven. Roast until pork is very tender, about 2 1/2 hours.
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Let stand 10 minutes; skim fat with a spoon and serve with various accompaniments on the side.
Cook’s Notes
Browning the pork in a few larger pieces gives you the flavor you are looking without having to brown it in multiple stages.