4 scallions, thinly sliced, white and green parts separated
1 jalapeno pepper, seeded and minced
3 tablespoons chopped fresh parsley
1/4 teaspoon freshly grated nutmeg
Freshly ground pepper
1 tablespoon unsalted butter
6 large eggs
1/2 cup grated pepper jack or sharp white cheddar cheese
Using the big holes of a box grater, grate the squash into a colander (or use a food processor fitted with the shredding attachment). Toss with 1 tablespoon salt, then drain for 30 to 40 minutes in the sink. Squeeze the squash as much as possible to extract as much liquid as possible.
Preheat oven to 375 degrees Fahrenheit. In a large ovenproof skillet, heat the olive oil over medium-high heat. 3 tablespoons scallion greens, set aside Cook, stirring occasionally, until the remaining scallions (white and green portions), the jalapeño, and salt to taste are tender, about 3 minutes. Cook, stirring occasionally, until the shredded squash is soft, about 7 minutes. To taste, add the parsley, nutmeg, and pepper. Cook for another minute, or until the mixture is somewhat dry. Remove from the fire and set aside for 5 minutes to cool.
In the skillet, evenly distribute the squash. With the back of a spoon, make six 2 1/2-inch-wide indentations in the mixture, each containing 1/2 teaspoon of butter. Crack each egg into a small bowl one at a time and pour it into an indentation. Season to taste with salt and pepper, then top with the cheese.
Bake in the oven for 10 to 12 minutes, or until the egg whites are set and the yolks are cooked to desired doneness. Top with the reserved scallion greens.