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In a Dutch oven or large, heavy pot with a lid, heat the olive oil over medium. Add the onion and saute until tender and translucent, about 4 minutes.
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While the onions cook, pull any tiny bones from the salmon (tweezers can be useful for this). Cut into four equal pieces. Set aside.
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When the onions are tender, add the coconut milk, water, curry paste, brown sugar, and fish sauce, and stir until the ingredients are blended and the sauce is smooth. Add the sweet potato, stir, and turn the heat to high. When the liquid boils, drop the heat until it simmers, put on the lid, and cook the sweet potato until it is just tender, about 8 minutes.
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Nestle the 4 salmon pieces, skin side up, down into sauce surrounded by the sweet potato. Return the lid to the pot and simmer for 3 minutes.
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Use a spatula to gently turn the salmon over. Add the zucchini to the pot, doing your best to nestle it into the sauce. Return the lid and simmer until the salmon is done (it will be opaque in color and easily pull apart, about 3 minutes more (the time will vary depending on the thickness of the fish and it will continue to cook a bit even once it’s off the heat). Scatter the spinach over the curry and cover again with the lid. The spinach will wilt quickly.
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