Preheat the oven to 375 degrees F. Butter a 2-quart baking dish.
Melt the butter in a large pot over medium heat. Add the mushrooms, 1 teaspoon salt and a couple turns of pepper and cook until the mushrooms are tender, about 4 minutes. Add the milk and heavy cream and heat until very warm.
Whisk together the polenta, sugar, baking powder, 1 teaspoon salt and a couple turns of pepper in a large bowl. Whisk the mushroom sauce into the polenta mixture until well combined and thick. Whisk the egg yolks in a small bowl. Whisk 1/2 cup of the hot polenta into the yolks, repeat twice, then pour the yolk mixture into the remaining polenta. Stir in the Cheddar.
Whisk the egg whites with a pinch of salt in a medium bowl until soft, glossy peaks form. Fold the egg whites into the polenta mixture in three additions.
Spoon the polenta mixture into the prepared baking dish. Bake until golden and puffed, 25 to 30 minutes. Serve immediately.
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