Melt the coconut oil in a large skillet over medium heat. Add the pepper and broccoli and sauté for 3 minutes.
Add the mushrooms and continue cooking until the vegetables are tender. Stir in the spinach and continue to cook until the spinach is wilted, about 1 minute. Remove from the heat and set aside.
In a food processor, combine the tofu, soy milk, nutritional yeast, arrowroot powder, mustard, garlic, sea salt, turmeric, and ground pepper and process until smooth.
Heat an 8-inch non-stick sauté pan over medium heat. When your pan is hot, pour a heaping ⅓ cup of batter into the centre of the pan. Using a spatula, spread the batter out into an even circle, covering the bottom of the pan. Cook the omelette until the top is dry, about 4 minutes. With the spatula, gently loosen the edges of the omelette from the pan and turn the heat down to medium low. Sprinkle 2 tablespoons of the grated cheese and ¼ of the vegetable mixture over the bottom half of the omelette and flip the top half over the filling. (Note: The top half will not completely cover your filling, and that’s okay.)
Continue cooking the omelette until the cheese melts, about 2 minutes. Slide the omelette onto a plate. Repeat with the remaining batter, cheese, and vegetable mixture.