In a medium bowl, whisk together the miso paste, mirin, sake, sugar, soy sauce, and sesame oil until smooth.
Place the salmon fillets in a shallow dish or a resealable plastic bag. Pour the miso mixture over the salmon, ensuring each fillet is well coated. Cover the dish or seal the bag and refrigerate for at least 2 hours, or up to overnight for a deeper flavor.
Preheat your grill to medium-high heat. If using a grill pan, heat it over medium heat on the stovetop.
Remove the salmon from the marinade and let any excess marinade 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 another 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 minute. Garnish with sliced green onions and serve with lemon wedges.