In a small saucepan, combine soy sauce, mirin, sake, brown sugar, ginger, and garlic. Bring to a gentle boil over medium heat, then reduce to a simmer. Stir continuously until the sauce thickens slightly, about 5-7 minutes. Remove from heat and set aside.

Place salmon fillets in a shallow dish and pour half of the teriyaki sauce over them. Let the salmon marinate for at least 15 minutes, turning once to coat evenly.

Preheat a non-stick skillet over medium heat and add sesame oil. Place the marinated salmon fillets skin-side down in the skillet. Cook for 4-5 minutes on each side, or until the salmon is cooked through and caramelized. Brush with additional teriyaki sauce while cooking. Remove from heat and let the salmon cool slightly before flaking it into bite-sized pieces.

Rinse the sushi rice under cold water until the water runs clear. In a medium saucepan, combine rinsed rice and water. Bring to a boil over medium-high heat. Reduce heat to low, cover, and let it simmer for 15 minutes. Remove from heat and let the rice sit, covered, for an additional 10 minutes. In a small bowl, mix rice vinegar, sugar, and salt. Gently fold this mixture into the cooked rice. Let the rice cool to room temperature before using.

In a bowl, combine diced avocados with lime juice. Season with salt and pepper to taste. Mash lightly if desired for a smoother texture. Set aside for layering.

Use a cylindrical mold or a small bowl to shape the stack. Start by pressing a layer of seasoned rice at the bottom. Add a layer of flaked teriyaki salmon, pressing gently. Follow with the mashed avocado mixture. Repeat the layers if desired for a taller stack. Carefully remove the mold or invert the bowl onto a serving plate. Garnish with sesame seeds, chopped green onions, and cucumber if using. Drizzle a bit of the remaining teriyaki sauce over the stack for extra flavor.