Place rice in a fine mesh strainer and rinse until the water runs clear. Place in a sauce pan with water and bring to a boil. Reduce the heat to a simmer, cover and cook until all water is removed (about 20 minutes). Set aside to cool.
In a small sauce pan, combine vinegar, sugar, olive oil and salt. Simmer over medium heat until sugar is dissolved. Allow to cool slightly and mix into the rice.
Place one sheet of nori onto a sushi rolling mat (if you do not have a sushi rolling mat, you can also use a clean kitchen towel). Spread ¼ of sushi rice onto the nori sheet, leaving about ¼ inch on each side.
Arrange 2 spears each of asparagus and carrots with a few pieces of cucumber and avocado in a line towards the center of the rice. Begin to roll the sushi by taking the edge closes to you and folding it onto of the veggies and continue rolling while squeezing to gently compress the roll together.
Continue with remaining sheets, rice and veggies.
Slice with a thin, sharp knife, sprinkle with toasted sesame seeds and serve immediately with soy sauce and Wasabi Horseradish Cream Sauce.