Preheat the oven to 350 degrees F (175 degrees C).
Heat 1/2 of the oil in a skillet over medium heat. Cook and stir 1/2 of the garlic and ginger until softened, 3 to 5 minutes. Remove from heat and cool for 10 minutes.
Grease a baking pan with the remaining oil. Season salmon with salt and pepper and place in the baking pan. Rub cooled garlic and ginger over the salmon.
Bake in the preheated oven for 12 minutes.
While the salmon is baking, prepare the hoisin-plum sauce. Melt butter in the skillet; cook and stir remaining garlic and ginger until softened, 3 to 5 minutes. Add rice vinegar and let it cook off. Add oyster sauce and chile paste; bring to a simmer. Stir in hoisin and plum sauces and bring back to a simmer. Pour in chicken broth to loosen up sauce to a lighter consistency. Add soy sauce.
Cover salmon with the sauce and continue baking until flesh flakes easily with a fork, 5 to 6 minutes more. Garnish with scallions before serving.