In a shallow bowl, toss together bread crumbs, five-spice powder, thyme, allspice, salt and cayenne. In a second shallow bowl, beat eggs with a fork. Turn fish fillets one at a time in crumbs, then dip them in beaten eggs to coat and then coat again with crumb mixture; place on a plate.
In a large skillet, heat 2 Tbsp. oil over medium heat until hot. Add half of fish and cook until golden and crisp on bottom, 2 to 3 minutes; turn and cook until crisp on other side, 2 to 3 minutes more. Transfer fish to a plate and keep in a warm oven while you repeat with remaining fish and 2 Tbsp. more oil.
In a large nonstick skillet, heat remaining 2 Tbsp. oil over medium-high heat. Slice bananas 1/3-inch thick and pan-fry until golden on both sides, about 4 minutes. Serve bananas and lime wedges with fish.
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