Rinse the sushi rice under cold water until the water runs clear. This removes excess starch and helps achieve a better texture.
In a rice cooker or pot, combine the rinsed rice and water. Cook according to the rice cooker instructions or bring to a boil, then reduce to low heat, cover, and simmer for 20 minutes.
Once cooked, remove the rice from heat and let it sit covered for 10 minutes. In a small bowl, mix rice vinegar, sugar, and salt until dissolved, then fold this mixture into the rice gently to season it.
Spread the seasoned rice onto a baking sheet, spreading it evenly, and allow it to cool to room temperature.
Once cool, press the rice firmly into a square or rectangular shape about 1/2 inch thick. Refrigerate for 30 minutes to firm up the rice.
Preheat a non-stick skillet over medium heat and add sesame oil.
Cut the chilled rice into bite-sized square or rectangular pieces. Dredge each piece in panko breadcrumbs, coating evenly.
In the hot skillet, place the coated rice pieces and fry for 2-3 minutes on each side, until golden brown and crispy. Remove and place on a paper towel to drain excess oil.
Top each crispy rice piece with a slice of salmon, avocado, and cucumber. You can also drizzle a little extra soy sauce over the top for added flavor.
Serve immediately, garnished with wasabi and pickled ginger on the side.