Put avocado pieces and lime juice into a food processor or blender and pulse until blended. Slowly add olive oil, pulsing, until you reach desired consistency of sauce. Add minced shallots (or green onions) and parsley, pulse just until combined. Remove to a bowl, add mustard, salt and pepper to taste.

Coat the bottom of a sauté pan with oil, heat on medium high until almost smoking. Season both sides of the salmon fillets with salt and pepper, carefully lay the salmon into the pan, skin side down. Cook the salmon until about medium doneness, about 3-4 minutes per side.