Heat 1 1/2 tablespoons vegetable oil in a large skillet over medium-high heat. Season salmon generously with salt and pepper, then place skin-side down in the pan. Cook for 4 minutes until the skin is crispy.

Flip the salmon and add 2 tablespoons butter, the minced ginger, and minced garlic to the pan. Tilt the pan and spoon the foaming butter over the salmon repeatedly for 3–4 minutes until it's cooked through (it should flake easily with a fork).

Drizzle 2 tablespoons soy sauce over the salmon and turn off the heat. Let it rest in the pan while you finish the rice.

In another large skillet or wok, heat the remaining 1 1/2 tablespoons vegetable oil over medium-high heat. Pour in the beaten eggs and scramble until just set, then push to the side.

Add the cooled rice, breaking up any clumps with your spatula. Stir-fry for 2–3 minutes until the rice is heated through and coated with oil.

Add the frozen peas and carrots, the remaining 1 tablespoon soy sauce, and the sesame oil. Toss constantly for 1–2 minutes until the vegetables are warm.

Divide the fried rice among plates and top each with a piece of salmon, spooning any pan juices over the top. Scatter chopped green onion over everything.