To make the salmon cakes: Heat 2 tablespoons of the peanut oil in a skillet over medium heat. Add the garlic, shallot, ginger, and chiles; saute for a few minutes to release the flavor. Remove from heat to cool slightly.
In a mixing bowl, combine the salmon, bread crumbs, cilantro, mayonnaise, lemon juice and egg white. Scrape the garlic/ginger oil into the salmon mixture; season with salt and pepper. Fold the ingredients together gently but thoroughly, taking care not to mash the salmon too much. Using your hands, form the mixture into 4 salmon cakes, they should be moist and just hold together.
Coat a non-stick skillet with the 2 tablespoons of peanut oil and bring it to a slight smoke over medium heat. Fry the salmon cakes until brown, about 4 minutes on each side, turning carefully with a spatula. Keep warm on a plate lined with paper towels.
To make the bok choy: Return the skillet to the heat and coat with the 2 tablespoons of peanut oil and heat until almost smoking. Split the piece of ginger open with a knife then whack it with the flat side of the knife. Lay the ginger pieces in the oil, cut-side down to let it start to perfume. Pan-fry the bok choy, cut-side down, for a couple of minutes to give it some color. Add the water and cook another minute to steam the bok choy; carefully remove it to a plate. Add the soy sauce, oyster sauce, lemon juice, and brown sugar. Cook and stir for 3 minutes until the sauce is the consistency of syrup.
Serve the salmon cakes with the bok choy, and drizzle the brown sauce over the whole thing. Garnish the dish with the toasted sesame seeds, cilantro, and green onion.