Special Equipment: a jumbo 6-cup muffin tin and jumbo muffin liners
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.
Cook the eggs: Put 6 cold eggs in a medium saucepan. Add enough water to cover by about 1 inch. Bring to a boil, cover, remove from the heat and let sit for 4 minutes. Transfer the eggs to a bowl of ice water. Gently crack the eggs so that they fracture all over and let them sit in the water for at least 20 minutes.
Gently peel the eggs under cold running water. (Older eggs are easier to peel.)
Make the muffin batter: Melt the 2 tablespoons of soft butter in a medium skillet over medium-high heat. Once the butter starts to foam and cook, add the onions and cook, stirring frequently, until golden brown, about 4 minutes. Add the ham and 1/4 teaspoon salt and continue to stir until browned in spots, about 2 minutes. Remove from the heat and stir in the thyme. Cool in the skillet.
Combine the flour, cornmeal, sugar, baking powder, baking soda, 1 1/2 teaspoons salt and 1/4 teaspoon pepper in a large bowl.
Whisk together the milk, sour cream, 4 tablespoons of the melted butter and the 3 remaining eggs in a separate medium bowl. 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).
Divide half of the batter evenly among the prepared muffin cups. Stand one egg in each cup, nestling it slightly in the batter. 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.
Lightly brush the top of the batter in the cups with the remaining 2 tablespoons of melted butter. Sprinkle with a little salt and pepper. Bake until the muffin tops are golden, 35 to 40 minutes. Let the muffins rest in the hot muffin tin for at least 10 minutes. Remove, cut each muffin in half and serve warm, or transfer to a rack to cool to room temperature, cut and serve.
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