Place the porcini mushrooms in a small bowl. Add ¾ cup very hot water and let soak until soft, about 20 minutes. Drain the porcini, squeeze out the water. Chop the mushrooms.

Heat the oil in a heavy medium skillet over medium heat. Add the shallot and sauté 1 minute. Add the baby bella mushrooms, sprinkle with salt and pepper and sauté until beginning to soften, about 3 minutes. Add the porcini and sauté 1 minute to blend the flavors. Mix in ½ teaspoons thyme. Taste and adjust the seasonings. Remove from the heat. Mix the cheeses together in a medium bowl.

Break 3 of the eggs into a small bowl. Sprinkle with a big pinch each of salt and pepper. Beat with a fork until the whites and yolks are thoroughly blended, about 35 strokes.

Place a nonstick skillet or omelet pan with a 7-inch-diameter bottom over medium-high heat. Add 1 tablespoon of the butter and let cook until melted and the foam almost subsides. Add the eggs all at once. Using a wooden spatula, stir along the bottom of the pan a few times until the eggs thicken, tilting the pan occasionally and lifting the edge of the eggs to allow uncooked egg to flow under the cooked portion. Cook only until the top is almost set but still very wet. This will take only a few seconds. Sprinkle half the cheese over the eggs, then spoon half the mushroom mixture in a line down the center of the eggs. Tilt the pan and fold one third of the eggs over the mushrooms, then fold the omelet over again. Let brown for a few seconds and then roll out the omelet onto a plate. Make a second omelet with the remaining eggs, butter, cheese and mushrooms. Sprinkle thyme over the omelets and serve right away.