Make onions: Combine vinegar, sugar, and salt in a small bowl with 1/2 cup water, then whisk until sugar is dissolved. Add onions and let sit 20 minutes.
Make sauce: Bring all ingredients to a boil in a small saucepan, then simmer for a few minutes until slightly thickened. Let cool.
Spread cooked rice on a baking sheet, then sprinkle with sesame seeds. Arrange a bamboo mat on a work surface with narrow end toward you. Lay one sheet nori, wide end toward you, on a bamboo mat. Moisten the far edge and lay second nori sheet on top of first; press to seal. Using wet hand, spread rice in a thin layer over just the first sheet of nori to cover it completely. Arrange salmon dice in a horizontal row about 1/3 way down first nori sheet and all the way to edges, then layer cucumbers and avocado on top. Drizzle with unagi sauce and sprinkle with potato chips, cilantro, and a sprinkle of furikake.
Using bamboo mat, roll filled nori into a tight roll about 3 in. thick. Continue rolling onto second nori sheet until burrito is covered with second sheet (this will help seal in moisture).
Wrap burrito in parchment paper to help hold it together, cut in half, and serve with remaining unagi sauce.
*Find in the international aisle of well-stocked grocery stores or at Asian markets.