In a small bowl, whisk together the miso paste, mirin, sake, sugar, soy sauce, and sesame oil until smooth and well combined.
Place the salmon fillets in a shallow dish and pour the miso marinade over them, ensuring they are well coated. Cover and refrigerate for at least 30 minutes, or up to 2 hours for a stronger flavor.
Preheat your grill or grill pan over medium-high heat.
Remove the salmon from the marinade, allowing any excess to drip off. Discard the remaining marinade.
Place the salmon fillets skin-side down on the grill. Cook for about 4-5 minutes, then carefully flip and grill for an additional 3-4 minutes, or until the salmon is cooked through and flakes easily with a fork.
Remove the salmon from the grill and let it rest for a couple of minutes.
Garnish with sliced green onions and serve with lemon wedges on the side.