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Bake the sweet potato on a baking sheet in a preheated 375°F oven for 1 hour, until easily pierced with a knife. Remove from the oven, cool, and peel. Purée in a food processor; you should end up with roughly 1 1/2 cups.
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Generously butter a 2-quart baking dish with straight sides. Place a rack in the center of the oven and preheat to 375°F.
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In a medium saucepan, heat the milk, sugar, salt, and cayenne until bubbles form around the edges. Whisk in the cornmeal and cook for about 2 minutes, stirring constantly, until the mixture thickens and a thin film coats the bottom of the pan. Remove the thickened mixture from the heat, and stir in the butter a tablespoon at a time.
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In a large bowl, whisk together the egg yolks and heavy cream until the yolks are fully incorporated and the cream is a uniform pale yellow.
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In the bowl of your mixer fitted with the whisk attachment, whip the egg whites until they form soft peaks.
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Once the cornmeal mixture has cooled for at least 5 minutes, whisk 1/3 of it into the egg yolks and cream. Follow up with the remaining cornmeal mixture to create a pour-able batter. Using a spatula, stir in the grated cheese, sweet potato purée, and poblano peppers. Finally, fold in half of the whipped egg whites. Continue with the remaining whites.
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Pour the batter into the prepared baking dish and bake for 35 to 40 minutes.
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Store any leftovers covered in the refrigerator for up to 5 days.