To prepare the salad, toss the chillies, cilantro leaves, mint leaves, shallots, snow peas and sliced lime leaves together in a bowl, then set aside.
To prepare the dressing, stir the brown and white sugars with 5 tablespoons water in a medium-sized saucepan over low heat, until the sugars have dissolved. Increase heat to medium, and continue to cook until lightly golden. Remove from heat, add the fish sauce and lime juice (be careful, it will splatter), and stir to combine.
To cook the fish, firstly mix the flour and salt together in a bowl. Add the fish and coat well.
Heat the oil in a heavy-bottomed (preferably nonstick) sauté pan just large enough to hold the salmon; it will be the right temperature when the oil forms ripples and barely begins to smoke.
For inch-thick fillets, sauté over high heat for 3 to 5 minutes on the first side, then turn with a spatula and sauté for 2 to 3 minutes on the second side, depending on how you like your salmon done; saute the salmon more on the first side (the presentation side) than the second, so that you end up with a savory golden crust or crisp skin.
Serve the fish topped with the spicy green salad. Spoon over the sweet and sour dressing, and serve with lime quarters.