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Special Equipment: a jumbo 6-cup muffin tin and jumbo muffin liners
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Adjust an oven rack to the middle position and preheat the oven to 350 degrees F. Line a jumbo 6-cup muffin tin with jumbo liners and generously grease with cooking spray.
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Cook the eggs: Put 6 cold eggs in a medium saucepan.
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Add enough water to cover by about 1 inch.
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Bring to a boil, cover, remove from the heat and let sit for 4 minutes.
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Transfer the eggs to a bowl of ice water.
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Gently crack the eggs so that they fracture all over and let them sit in the water for at least 20 minutes.
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Gently peel the eggs under cold running water.
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(Older eggs are easier to peel.)
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Make the muffin batter: Melt the 2 tablespoons of soft butter in a medium skillet over medium-high heat.
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Once the butter starts to foam and cook, add the onions and cook, stirring frequently, until golden brown, about 4 minutes.
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Add the ham and 1/4 teaspoon salt and continue to stir until browned in spots, about 2 minutes.
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Remove from the heat and stir in the thyme.
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Cool in the skillet.
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Combine the flour, cornmeal, sugar, baking powder, baking soda, 1 1/2 teaspoons salt and 1/4 teaspoon pepper in a large bowl.
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Whisk together the milk, sour cream, 4 tablespoons of the melted butter and the 3 remaining eggs in a separate small bowl.
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Fold the egg mixture, onion mixture, Cheddar and Parmesan into the dry ingredients until just combined (don't fret if there are lumps; it means your batter isn't over-mixed).
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Divide half of the batter evenly among the prepared muffin cups.
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Stand one egg in each cup, nestling it slightly in the batter.
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Spoon the remaining batter on top of the eggs, using the back of your spoon to spread the batter to make sure each egg is completely covered.
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Lightly brush the top of the batter in the cups with the remaining 2 tablespoons of melted butter.
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Sprinkle with a little salt and pepper.
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Bake until the muffin tops are golden, 35 to 40 minutes.
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Let the muffins rest in the hot muffin tin for at least 10 minutes.
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Remove, cut each muffin in half and serve warm, or transfer to a rack to cool to room temperature, cut and serve.