For The Jalapeno Slaw
Whisk 1 1/2 teaspoons of the vinegar, the mustard, salt, and pepper in a large bowl.
Slowly drizzle in olive oil, whisking constantly, until vinaigrette is emulsified.
Pour the remaining vinegar over the onion in a bowl; let sit 5 minutes, then drain, discarding vinegar.
Add onion, shredded cabbage, jalapeno peppers, salt and pepper to the bowl of the reserved vinaigrette. Stir well so that the dressing coats the ingredients. Cover the slaw with plastic wrap and chill until ready to use.
For The Buttermilk Chicken Burgers
Add salt and pepper and chicken breasts to buttermilk. Stir well, cover with plastic wrap and chill for 1 hour in the refrigerator.
In the meantime, whisk flour, chile powder, salt, and pepper in a shallow bowl.
Heat 2" canola oil in a large saucepan until a deep-fry thermometer reads 325°.
Working with one chicken breast at a time, dip chicken in flour mixture, shaking off excess. Then dip in buttermilk and, once again, in flour.
Fry chicken, flipping once, until golden and cooked through, about 8 minutes, or until an instant-read thermometer inserted into the thickest part of the breast reads 165°. Transfer chicken to paper towels to drain; season with salt.
To assemble sandwiches, place chicken on bottoms of rolls; top with coleslaw and and then add the top bun.