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Line a 13x18-inch rimmed baking sheet with aluminum foil and drizzle with 3 tablespoons olive oil. Place the pan in the oven while it preheats. Preheat the oven to 425 degrees F (220 degrees C).
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Stir together flour, onion powder, salt, garlic powder, paprika, and pepper in a shallow dish. Place beaten eggs in second shallow dish. Combine French-fried onions and breadcrumbs in a third shallow dish.
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Dip each chicken tender first into the flour mixture, then the egg mixture, and finally the onion–breadcrumb mixture until fully coated; set aside.
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Carefully remove the baking sheet from the oven and swirl to coat with the hot oil. Arrange coated chicken pieces onto the prepared baking sheet.
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Bake in the preheated oven for 8 minutes. Flip and continue baking until the chicken is golden brown and cooked through, about 5 to 6 minutes more, or until an instant-read thermometer inserted into the thickest part reads at least 165 degrees F (74 degrees C).
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Meanwhile, whisk together mayonnaise, sour cream, horseradish sauce, ketchup, white vinegar, onion powder, garlic powder, smoked paprika, salt, pepper, and cayenne for the dipping sauce in a bowl until smooth.
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Serve chicken with onion dipping sauce.