Sauté the aromatics: Heat a large non-stick pan over medium heat. Add the diced onion and 2 tablespoons of water. Cook for 3 to 4 minutes until the onion softens and turns translucent. Add more water a splash at a time if it starts to stick.
Cook the veggies: Add the thinly sliced cabbage, diced mushrooms, grated courgette, and grated carrot to the pan. Cook for 5 to 6 minutes, stirring frequently, until all the vegetables are tender and their moisture has mostly evaporated. Remove from heat and set aside to cool slightly.
Whisk the batter: In a large mixing bowl, whisk together the chickpea flour, nutritional yeast, garlic powder, baking soda, salt, and black pepper. Gradually pour in the ¾ cup of water, whisking constantly until you have a smooth, lump-free batter.
Combine: Fold the cooked vegetable mixture into the chickpea batter and stir well until all the vegetables are evenly coated.
Cook the omelettes: Wipe down your non-stick pan and return it to medium-low heat. Pour half of the batter into the pan, spreading it gently into a neat circle. Cover with a lid and cook for 4 to 5 minutes until the edges look dry and bubbles form on top. Carefully flip and cook the other side for another 2 to 3 minutes until golden brown. Repeat with the remaining batter.
Make the sauce: While the omelettes are cooking, combine the Greek-style soy yogurt, garlic powder, apple cider vinegar, a pinch of salt, and chopped fresh parsley in a small bowl. Stir until smooth.
Plate and serve: Place the hot omelettes on plates, drizzle generously with the garlic-herb yogurt sauce, and sprinkle with extra fresh parsley. Serve immediately with a side of fresh or lightly blistered cherry tomatoes.