Position a rack in the top third of the oven and heat the oven to 425°F.
Heat the butter or oil in a 10-inch skillet over medium heat. Add the bacon and shallots and cook, stirring, until they begin to brown, 8 to 10 minutes. Stir in the sugar and cook until the bacon and shallots turn deep golden brown, about 4 minutes. Remove from the heat and add the vinegar. Stir in the sun-dried tomatoes; set aside to cool.
Put the cornstarch in a medium bowl. Whisking steadily, slowly pour in 1/2 cup of the milk, mixing until quite smooth. Whisk in the whole eggs and egg yolks, mixing again until smooth, and then gradually whisk in the rest of the milk, the cream, salt, and cayenne.
Butter or oil a 2-quart, 7×11-inch Pyrex dish or an 8×11-inch ceramic gratin dish. Spread the ham and mushroom mixture over the bottom of the pan. Scatter on the Brie and parsley.
Whisk the batter in case it has settled and then pour it into the baking dish. Bake until the top is deep golden and the custard is set (insert a pick in the center to check; it should be softly set, neither liquid nor firm), 25 to 30 minutes. Let stand for 30 minutes before serving.
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