In a saucepan, combine the vinegar, water, sugar, and salt. Heat until sugar dissolves, about 5 minutes.
Pour over julienned carrots and let sit for 30 minutes.
To make the peanut sauce, whisk all the sauce ingredients in a saucepan. Simmer for 3 minutes until thickened.
Set up breading station: One bowl with flour, a second bowl with beaten eggs, and a third bowl mixture of crushed peanuts and breadcrumbs.
Dredge the chicken breasts in flour, then eggs, then the peanut-panko mixture.
Heat ¼-inch oil in a large pan.
Pan-fry the chicken for 3 to 4 minutes per side, until golden and cooked through.
Toast the brioche buns.
Assemble the sandwich in the following order: bottom bun, 1 tablespoon peanut sauce, chicken, cucumber ribbons, quick-pickled carrots, cilantro, another 1 tablespoon of peanut sauce, top bun.