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Preheat oven to 350°F. Coat a 9-by-13-inch baking dish with cooking spray.
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Bring a medium pot of water to a boil. Add yuca, cover and cook until completely tender, 30 to 35 minutes. Drain and remove any fibrous strings from the yuca. Mash with a potato masher until lump-free, then stir in egg and milk until completely creamy and smooth.
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Meanwhile, heat oil in a large skillet over medium-high heat. Add pepper and onion; cook, stirring, for 2 to 3 minutes. Add chicken, lemon juice and paprika; cook, stirring occasionally, until the chicken is no longer pink, about 5 minutes. Add marinara and salt and cook until warmed through, about 3 minutes.
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Layer half of the yuca mash in the bottom of the prepared baking dish. Add the chicken mixture and top with 1/4 cup Cheddar and 1/4 cup mozzarella. Evenly top with the remaining yuca mash and sprinkle with the remaining 1/4 cup each Cheddar and mozzarella. Bake until the cheese is melted and the edges are golden, about 30 minutes. Let stand for 5 minutes before cutting into 6 pieces.