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1
Using a whisk or fork, beat the egg and milk together in a small bowl or a deep pie plate; set aside.
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2
Spread 1/2 teaspoon of the mayonnaise on each slice of bread.
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3
Arrange the ham and cheese on 1 slice of bread.
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4
Top with the other slice of bread, mayonnaise side down.
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5
Heat a large cast-iron skillet over medium heat until very hot.
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6
Melt 1 teaspoon of the butter in the hot skillet.
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7
Holding the sandwich together carefully between your fingers, dip both sides of the sandwich into the egg batter.
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8
Let any excess drip off.
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9
Lay one side of the dipped sandwich in the skillet.
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10
Cook for 3 to 4 minutes, or until golden brown on the bottom.
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11
Using a spatula, flip the sandwich at the same time that you add the remaining teaspoon of butter to the skillet.
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12
Lift the sandwich to allow the butter to run underneath as it melts.
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13
Brown the sandwich on the second side for 4 to 5 minutes.
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14
Cut the sandwich in half or quarters and serve immediately.
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15
For croque madame, proceed with the above recipe.
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16
Keep the sandwich warm on a plate while you melt 1 teaspoon butter over low heat in the same skillet until sizzling.
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17
Crack an egg into the skillet and fry it, flipping once, until it is fairly firm.
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18
Add a little salt and pepper to taste.
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19
Arrange the fried egg on top of the sandwich, cut in half, and serve.
