In a small bowl, whisk together Mayonnaise (1/2 cup), Distilled White Vinegar (2 teaspoon), Granulated Sugar (1 teaspoon), Wasabi Paste (to taste), Salt and Pepper (to taste). Cover and keep chilled until ready to use.
When ready to assemble sliders, toss Green Cabbage (2 cup), Scallion (1/4 cup), and Carrot (1/4 cup) with dressing.
Slice Boneless, Skinless Chicken Breast (12 ounce) in half horizontally and then cut each half into half again, getting 4 pieces (each being about 1.5 ounces) out of each breast. Place the chicken pieces between two pieces of parchment or wax paper and pound with a rolling pin or flat side of a mallet to 1/4-inch thick.
Using 3 shallow bowls, to the first, add the All-Purpose Flour (1/2 cup), Salt and Pepper (to taste), and whisk to combine. To the second, add Egg (2) and Water (1 tablespoon) and whisk to combine. To the third, add the Breadcrumbs (1 1/2 cup).
Add several chicken pieces to the flour bowl and flip around to coat. Shake off excess and add pieces to egg bowl. Using a fork, flip to coat with egg and place in bread crumbs, turning each piece and patting gently to adhere crumbs. Place breaded portions onto a clean plate or wire rack and repeat breading procedure until all pieces are done.
Add enough Canola Oil (to taste) to a large skillet to just start to cover the bottom. Heat over medium high until hot. Add breaded chicken to pan, being careful not to overcrowd pan. Cook for approximately 2 minutes per side until golden.
Remove from pan to a paper towel lined tray to dab off any excess grease and then to a clean plate. Keep warm in a low 225 degrees F (110 degrees C) oven while remaining cutlets are fried. Wipe out pan with a paper towel to remove excess crumbs that may start to burn prior to frying more cutlets.
Slice Slider Buns (8) in half and brush lightly with Salted Butter (to taste). Wipe out skillet with a paper towel and place slider buns buttered side down in pan and heat over medium until starting to brown and become warm. About 2 minutes.
To each slider bun, add a chicken cutlet and top with a generous amount of wasabi slaw. Secure each slider with a long toothpick if desired and serve immediately.