Peel the cucumber and avocado. Slice into long, thin strips. Set aside.
Make the spicy mayo: combine mayonnaise and sriracha in a small bowl. Set aside.
Lay the bamboo mat on a flat surface. Cover the bamboo mat with a layer of plastic wrap. Press 1 cup of cooked rice down into a rectangular shape, leaving a perimeter of bamboo mat showing.
Cover the rice with another piece of plastic wrap. Use a rolling pin to smooth and flatten the rice. Remove the plastic wrap and lay a piece of nori on top of the rice (shiny side down).
Arrange strips of cucumber, avocado and chicken tempura in a line across the center of the nori sheet.
Starting from the bottom, holding on to the bamboo roller, fold the roll up and over the fillings. Press it down firmly. Rotate the roll 90 degrees and repeat the same action, folding the roll up and over the fillings. Press firmly.
Slowly release the bamboo map and unwrap the sushi roll from the plastic wrap. On a cutting board, use a wet knife to gently slice the roll into 8 pieces.
Drizzle with eel sauce and spicy mayo. Garnish with toasted sesame seeds or furikake, if desired.