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Preheat oven to 425°F. Melt 2 tablespoons butter in a saucepan over medium heat until foamy. Add flour and cook, stirring, until pale and begins to foam, 1 to 2 minutes. Gradually whisk in milk. Cook, stirring often, until mixture is a pudding-like thickness, 14 to 16 minutes. Remove from heat and whisk in mustard and nutmeg. Season with salt.
Place 4 slices bread on a rimmed baking sheet. Top with half of the béchamel, dividing evenly. Top with ham and half of cheese. Top with remaining bread, béchamel, and cheese, dividing evenly. Bake until cheese is golden brown and bread is crisp on bottom, 12 to 15 minutes.
Melt remaining 1 tablespoon butter in a nonstick skillet over medium heat. Crack eggs into skillet, making sure to keep them separate. Cook until whites are set but yolks are still runny, 3 to 5 minutes. Season with salt and pepper. Top sandwiches with eggs and sprinkle with parsley.