Heat 1 tablespoon olive oil in a medium saucepan over medium-high heat. Add the shallot, serrano, 1/4 teaspoon curry powder and a pinch each of salt and pepper. Cook, stirring, until the shallot is lightly browned, 2 to 3 minutes. Stir in the mango, raisins, 2 tablespoons vinegar and 1 cup water. Reduce the heat to medium low and simmer, stirring and lightly mashing the mango chunks, until the chutney thickens, 10 to 15 minutes.
Meanwhile, rub the top and sides of the salmon with the remaining 1 teaspoon curry powder; season with salt and pepper. Lay the salmon skin-side down in a cold nonstick skillet. Place the skillet over medium heat and cook until the salmon skin is crisp and the fish is almost cooked through, 10 to 12 minutes. Flip and cook 1 to 2 more minutes.
Heat the remaining 1 tablespoon olive oil in a large skillet over medium-high heat. Add the chard stems, garlic and a pinch each of salt and pepper. Cook, stirring, until crisp-tender, about 3 minutes. Add the chard leaves and cook, stirring, until tender, 3 to 5 minutes. Add a few tablespoons of water if the pan is dry.
Stir the mint and 1/2 teaspoon vinegar into the mango chutney. Season with salt and up to 1/2 teaspoon more vinegar as needed. Divide the chard and salmon among plates. Spoon the mango chutney on top of the fish; sprinkle with more mint.