Heat the vegetable oil in a large non-stick frying pan over moderate heat. Cook the salmon fillets (2 pieces at a time), until they're golden and just done, 3 to 4 minutes on each side depending on the thickness. About 2 minutes before the salmon is done, add the garlic slices to the oil to flavor the fish. (If the fillets are thick, cover and cook a few minutes longer.) Remove the salmon and garlic and drain on paper towels.
Combine the cucumber with the basil in a small bowl. Place a small amount of the cucumber mixture on individual plates. Arrange the salmon fillet on top and garnish with the cucumber mixture and sprinkle a few pieces of the cooked garlic on top. Drizzle with 2 tablespoons of the Ginger-Lime Sauce, top with peanuts and serve immediately.
Combine the garlic, chiles, sugar, ginger, fish sauce, lime juice, and water in a small bowl and stir well. Let the flavors develop by setting the sauce aside for 30 minutes before serving.