Place the flour in a shallow bowl and season generously with salt and freshly ground black pepper.
In a second shallow bowl, whisk the egg with 1 tablespoon water until combined.
In a third shallow dish, mix the panko with the white and black sesame seeds until evenly distributed.
Dredge each chicken tender in the seasoned flour, shaking off excess, then dip into the egg mixture, and press into the panko-sesame mixture to coat evenly on all sides; repeat with remaining tenders.
Heat a large non-stick skillet over medium-high and pour in vegetable oil to about a 1/4-inch depth; heat until shimmering.
Working in batches so the tenders do not touch, place coated tenders in the hot oil and cook 3–4 minutes per side, until golden brown and an instant-read thermometer inserted into the thickest part reads 165°F (74°C).
Transfer cooked tenders to a wire rack set over a baking sheet and keep warm in a 175°F (80°C) oven while you finish frying the rest; add more oil to the pan if needed between batches.
To make the dipping sauce, whisk together mayonnaise, minced garlic, lime juice, sriracha, soy sauce, and sesame oil until smooth.
If using the sauce as a dressing, whisk in 1–2 teaspoons water as needed to reach a drizzle-able consistency; otherwise serve as-is for dipping.