Step 1: Preparing the Teriyaki Sauce

1. In a small saucepan, combine the soy sauce, mirin, sake, brown sugar, minced garlic, and grated ginger.
2. Place the saucepan over medium heat and bring the mixture to a gentle boil, stirring occasionally until the sugar dissolves.
3. Reduce the heat to low and let it simmer for about 5 minutes until the sauce thickens slightly. Remove from heat and set aside.

Step 2: Marinating the Salmon

1. Place the salmon fillets in a shallow dish or resealable plastic bag.
2. Pour half of the prepared teriyaki sauce over the salmon, ensuring each fillet is well-coated.
3. Cover the dish or seal the bag and let the salmon marinate in the refrigerator for at least 20 minutes. (For a deeper flavor, marinate for up to an hour.)

Step 3: Cooking the Salmon

1. Heat the vegetable oil in a large skillet over medium-high heat.
2. Remove the salmon from the marinade, allowing any excess to drip off, and place the fillets skin-side down in the skillet.
3. Cook for about 4 minutes, then carefully flip the fillets and cook for an additional 3-4 minutes, or until the salmon is cooked through and flakes easily with a fork.

Step 4: Glazing and Serve

1. While the salmon is cooking, reheat the remaining teriyaki sauce if necessary.
2. Once the salmon is done, brush it generously with the warmed teriyaki sauce.
3. Transfer the salmon to serving plates, sprinkle with sesame seeds and chopped green onions, and serve immediately.