In a large bowl, combine soy sauce, mirin, sake, brown sugar, ginger, and garlic. Stir together until well-blended and sugar is dissolved.
Add salmon fillets and marinate for about 30 minutes. Drain salmon well and pat dry. Reserve marinade and set aside.
In a wide pan over medium-high heat, heat oil, swirling to cover the bottom of the pan.
Add the salmon in a single layer and cook for about 2 to 3 minutes.
Add the reserved marinade and cook for another 1 to 2 minutes or until salmon fillets are opaque halfway up the sides.
Using a spatula, gently turn over the fillets on the other side. Continue to cook, basting regularly with the sauce, for about 3 to 4 minutes or just until opaque and begins to easily flake with a fork,
Remove from pan and transfer to a serving platter. Garnish with toasted sesame seeds and chopped green onions, if desired.