If necessary, remove the tenderloins from the chicken and reserve for another use. Lightly pound the breasts between two sheets of plastic wrap to a uniform thickness of about 1/2-inch, then cut them crosswise into 2 or 3 bun-sized pieces (as close to 2-1/2-inches square as possible). Season on both sides with salt and pepper and set aside.
Next, prepare the slaw. Place the cabbage, carrot and cilantro in a large bowl. In a separate bowl, whisk together the oil, garlic, lime juice, Thai chili sauce and sriracha. Add to the cabbage and toss to combine. Season to taste with salt and pepper, add more sriracha if desired and set aside for the flavor to develop while you cook the chicken.
Whisk the egg and milk together in a shallow dish (a pie plate works well) and add the breadcrumbs to another. Dip the chicken breasts in the egg, allowing the excess to drip off, then press each piece into the crumbs to get an even coating on both sides. Set aside in a single layer.
Film the bottom of a large pan with vegetable oil and heat to medium-high. Working in batches, add the chicken and cook until golden brown and cooked through, 3 to 4 minutes per side. Transfer to a platter, cover to keep warm, and repeat with the remaining chicken. Note: You may want to wipe out the pan between batches and start with fresh oil.
To assemble the sandwiches, place a piece of chicken on each roll and brush with Thai chili sauce. Top with cilantro leaves and a spoonful of slaw. Serve immediately.