Heat oven to 400° F with a rack in the top position.
Heat the oil in a large ovenproof skillet over medium-high heat. Add the kale and cook, stirring, until wilted, 1 to 2 minutes. Add the marinara and bring to a simmer. One at a time, break each egg into a small dish, then slide it onto the sauce, spacing evenly. Season with ½ teaspoon each salt and pepper. Transfer to oven. Bake until the whites are just set, 10 to 12 minutes.
Top with the Feta and serve with the flatbread.
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