Bring a medium pot of salted water to a boil. Fill a large bowl with ice and water.
Add the spinach to the boiling water, stir to wilt for a few seconds and use a slotted spoon to transfer the spinach to the ice water bath. Drain the water from the spinach and transfer to a clean kitchen towel, twisting to wring out all the moisture. Separate the clumps as much as possible and set aside.
Preheat the oven to 400 degrees F.
Place a medium skillet over medium-high heat and add the butter. When the butter has melted, add the shallots and cook, stirring, until soft and browned slightly, about 2 minutes. Add the garlic and ham and cook 1 minute more. Add the cream and whisk in the mustard, nutmeg, a pinch of salt and a few grinds of pepper. Bring to a simmer, then whisk in the Parmesan. Simmer until thickened slightly, 2 to 3 minutes, then stir in the spinach and turn off the heat.
Put four 8-ounce ramekins on a baking sheet and divide the cream sauce among them. Use a spoon to push some of the filling to the edges and make some room in the middle. Crack an egg into the center of each ramekin and sprinkle the top of each egg with salt and pepper. Top each ramekin with 1 tablespoon shredded mozzarella.
Bake until the yolks are cooked to your desired doneness and the tops are bubbling and starting to brown, 15 to 20 minutes. Sprinkle with parsley and serve.
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