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Preheat oven to 400°F with rack in middle.
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Butter a 1-qt shallow baking dish or a 9-inch pie plate. Arrange bread in 1 layer in dish.
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Whisk together milk, 1 egg, and 1/4 tsp each of salt and pepper. Pour over bread. Sprinkle with cheese and press lightly to help bread absorb custard. Let stand until absorbed, 15 to 30 minutes.
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Dot with butter and bake, uncovered, until puffed and golden, 20 to 25 minutes.
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Bring 2 inches water to a boil with a few drops of vinegar in a small deep skillet or saucepan. Break each remaining egg into a cup and slide into water, spacing eggs apart. Poach at a bare simmer to desired doneness (we prefer firm whites with runny yolks, 2 to 3 minutes).
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Meanwhile, toss greens with remaining 2 tsp vinegar, oil, and 1/8 tsp salt.
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Divide pain perdu between 2 plates and top with salad. Lift eggs out of poaching liquid 1 at a time with a slotted spoon and place on top of salad. Lightly season eggs with salt and pepper.
