How to Make It
Preheat the oven to 250°. Bring a large saucepan of salted water to a boil. Prepare a bowl of ice water. Add all of the herbs to the boiling water and cook for 4 minutes. Using a slotted spoon, transfer the herbs to the ice water to cool completely. Transfer the herbs to a kitchen towel and wring dry. Transfer the herbs to a blender and add the 3/4 cup of water. Blend at high speed until pureed. Season with salt.
In a medium skillet, melt 1 tablespoon of the butter. Add the minced shallot and cook over moderate heat until softened, about 2 minutes. Add the chanterelles, season with salt and cook, stirring, until tender, 8 minutes. Remove from the heat, stir in the crème fraîche and keep warm.
In a 6- or 8-inch nonstick, ovenproof skillet, melt 1 teaspoon of the butter in the oil over moderate heat. In a small bowl, beat 1 egg with a pinch of salt until the egg foams. Add the egg to the skillet and cook over low heat until it starts to set, about 10 seconds. Transfer the skillet to the oven and cook until the egg is just set, about 1 minute. Slide the egg onto a plate and spread a rounded tablespoon of chanterelles down the center. Fold both sides of the omelet over the mushrooms and roll the omelet over to make a neat package. Spoon a little of the herb sauce alongside and serve immediately. Repeat the process 3 times with the remaining eggs, chanterelles and herb sauce, using 1 teaspoon of butter for each omelet.