TO MAKE THE FILLING: In a medium skillet, heat the butter over medium heat until melted. Add the shallot and saute until soft; about 2 minutes. Add the mushrooms and saute another 2 minutes. Add the zucchini and continue to saute for 2 minutes.
Add the tomato and cook over high heat for 2 minutes to evaporate any excess liquid from the tomato. Add the salt, pepper, and basil and mix well. Taste for seasoning. Cover to keep warm.
TO PREPARE EACH OMELETTE: Whisk 2 to 3 eggs with ta pinch of salt, a pinch of pepper, and 1/2 teaspoon club soda until smooth. Melt 1 tablespoon butter in an 8-inch omelette pan or skillet over medium heat until it begins to sizzle. Pour in the egg mixture and stir it in the center of the pan with the flat side of a fork. With the prongs of the fork, lift the edges on the omelette so any uncooked mixture runs to the edge of the pan. Vigorously slide the pan back and forth over the heat until the omelette begins to slip around freely. When the omelette is lightly cooked but still creamy in the center, spoon about 2 tablespoons of the filling over the half of the omelette closer to the pan's handle. Sprinkle 1 tablespoon of the parmesan cheese over the filling. Quickly jerk the pan toward you with the handle so that the other half of the omelette flips over on top of the filling. Slide the folded omelette onto a serving dish.
Serve immediately or keep warm in a low oven while preparing the remaining omelettes.
ADVANCE PREPARATION: The filling may be prepared 1 day in advance without the basil and refrigerated. Remove it from the refrigerator 1/2 hour before using, add the basil and gently reheat.