We make this with beautiful fresh eggs we get from our friends
Make the Parmesan béchamel sauce: Melt the butter in a saucepan over medium heat. Add the flour and stir constantly for a few minutes to cook the flour; do not allow it to take on any color. Gradually pour in the milk as you stir and whisk out any lumps. Continue to stir until the mixture reaches a boil and thickens, then add the Parmesan cheese and nutmeg. Season with salt and pepper; set aside and keep warm.
Set a skillet over high heat and add the butter. When the butter melts, add the shallots and sauté for 20 seconds, then add the mushrooms. Sauté until the moisture has evaporated and the mushrooms have developed a nice golden color, about 5 minutes; season with salt and pepper.
Set a gas grill to high heat. If using a charcoal grill, pile the briquettes higher on one side and lower on the other for a hot side and a medium side. Oil the grill grates with olive oil. Roll out the pizza dough to a ¼-inch thickness and place directly on the grill over the hottest part of the flame. Grill on both sides until slightly charred, then transfer to a cutting board. While the crust cooks, grill the onions until soft and slightly charred, about 5 minutes.
Spoon the béchamel onto the pizza crust and spread evenly. Dot with the mushrooms and arrange the spring onions on top. Return the pizza to the grill, reducing the heat to medium or placing it over the less hot part of the fire, and gently crack the eggs on top. Cover the grill and cook until the eggs have just set, 5 to 7 minutes. Remove from the heat and sprinkle with the Parmesan and flowers, if using.
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