Position a rack in the center of the oven and heat the oven to 375°F. Lightly butter a 6-cup baking dish, preferably oval.Bring the milk and 1/2 tsp. salt to a simmer in a 4-quart saucepan over medium heat. Whisk in the polenta. Continue whisking until thick, 3 to 5 minutes. Transfer to a large bowl and set aside.
Melt 2 Tbs. of the butter in a 12-inch skillet over medium-low heat. Add the onion and a pinch of salt and cook, stirring occasionally, until golden and soft, about 15 minutes. Add the ham and stir often until warmed through, about 2 minutes.
Stir the ham mixture, cheeses, egg yolks, basil, oregano, thyme, hot pepper sauce, 1/2 tsp. salt, and 1/2 tsp. pepper into the polenta; mix well.
In a medium bowl, beat the egg whites with an electric mixer at high speed until soft peaks form. Fold the beaten egg whites into the polenta and then spread the mixture in the prepared baking pan. Cut the remaining 2 Tbs. butter into pieces and dot on top.
Bake until brown and puffed, 30 to 35 minutes. Cool on a wire rack for 10 minutes before serving with a large spoon.
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