Make the rice: Put the rice in a small saucepan and add 1 1/4 cups water. Bring to a boil, then cover, reduce the heat to low and cook until the water is absorbed, about 15 minutes. Remove from the heat and let stand, covered, 10 minutes. Meanwhile, whisk the rice vinegar, sugar and 1 teaspoon salt in a large bowl. Gently stir in the sesame seeds and cooked rice with a rubber spatula and let cool to room temperature. (The rice can be made up to 2 hours ahead; cover with a damp cloth.)
Meanwhile, make the sauce: Combine the mayonnaise, lemon juice, Sriracha, soy sauce, sesame oil and ginger in a medium bowl. Cover and refrigerate until ready to use.
Make the tuna: Rub the tuna with 1/2 teaspoon each salt and pepper. Heat a nonstick skillet over medium-high heat until hot. Add the vegetable oil, then add the tuna and sear until golden brown, about 2 minutes per side. Transfer to a plate and refrigerate until ready to use, or up to 1 hour.
Assemble the rolls: Toast the seaweed by waving it directly over a gas or electric burner over medium heat. Cut each sheet in half with kitchen shears to make 12 rectangles. Thinly slice the tuna and season with salt. Lay a seaweed rectangle on a work surface with a long side facing you. Press some rice onto the left side of the sheet using moistened fingers, leaving a 1-inch border on the left. Top with 1 slice each of tuna and avocado, a spoonful of sauce, and some cucumber, carrot and chives. Starting from the bottom left corner, tightly roll into a cone shape, moistening the edge of the seaweed to seal. Repeat with the remaining ingredients. Serve with pickled ginger and more sauce.