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. Cook the salmon fillets skin-side down for 4-5 minutes on each side, or until cooked through and caramelized. Brush with additional teriyaki sauce while cooking. Remove from heat and let cool slightly before flaking into bite-sized pieces.

Rinse 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 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 and let 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 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 some of the remaining teriyaki sauce over the stack for extra flavor.