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1
In the bowl of a stand mixer, whisk together the milk, sugar and yeast, then bloom until foamy, 10 minutes.
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2
In a medium bowl, whisk together the flour and the 1 tablespoon salt. Once foamy, add the flour mixture, 4 tablespoons of the melted butter and the egg to the milk mixture. Using the dough hook attachment, knead on medium speed until a smooth dough forms, 4 to 5 minutes. Transfer to a bowl greased with 1 tablespoon of the vegetable oil, cover with plastic wrap and let proof in a warm place until doubled in size, about 1 hour.
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3
After the first proof, punch down the dough and transfer to a sheet pan greased with another 1 tablespoon vegetable oil and sprinkled with 1 tablespoon of the semolina. Using a rolling pin, roll out the dough into a 12-inch circle. Brush with the remaining 1 tablespoon vegetable oil, sprinkle the remaining 1 tablespoon semolina flour and cover with parchment paper. Allow the muffin to proof for 30 minutes more.
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4
Preheat the oven to 425u00b0 with a clean sheet pan on the middle rack. Once heated carefully remove the hot sheet pan and brush with 1 tablespoon of the melted butter. Carefully transfer the muffin to the hot pan and cook until puffed and beginning to get color on the bottom, 10 to 12 minutes. Remove the hot pan from the oven and place the sheet pan used for proofing over the muffin. Invert the pans and return the muffin to the oven to cook until completely golden and has reached an internal temperature of 190u00b0, 10 to 12 minutes more. Remove from the oven and let cool completely.
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5
Use a carving fork to open the muffin and place each half on one of the used sheet pans, nooks-and-crannies-side up. Brush each half with 2 tablespoons of the melted butter and season each with salt. Place in the oven an toast until golden, 10 to 12 minutes.
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6
Meanwhile, in a 12-inch nonstick skillet, heat 1 tablespoon of the melted butter over medium-high heat. Brush 2 of the ham steaks with half the maple syrup and pan sear until golden and warmed through, 3 to 4 minutes per side. Repeat with another 1 tablespoon of butter and the remaining ham and maple syrup. Keep warm.
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7
In a 14-inch nonstick skillet, heat the remaining 1 tablespoon of melted butter over medium-high heat. Add the eggs and using a fork, break each yolk, but do not scramble them. Season with salt and cook, covered, until just set, 3 to 4 minutes.
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8
As soon as the muffin halves come out of the oven, shingle the American cheese on the bottom half, then top with the ham steak and the eggs. Place the top half of the muffin over the eggs, then serve.
