Rinse the spinach and let it drain well. Halve the cherry tomatoes and finely mince the shallot. Break the eggs into a bowl, add the milk, and season with salt and freshly ground black pepper. Whisk everything together vigorously until smooth and airy.
Heat the coconut oil or olive oil in a frying pan over medium heat. Sauté the shallot until translucent. Add the spinach in batches and cook, stirring, until the leaves have wilted. Add the cherry tomatoes and cook for another minute. Then distribute the vegetables evenly over the bottom of the pan.
Pour the egg mixture over the vegetables and immediately reduce the heat. Let the eggs set slowly. Push the cooked edges slightly inward with a spatula so the liquid egg can flow to the hot bottom. Wait until the top of the omelet is almost dry but still has a slight shine.
Fold the omelet in half with the spatula and let it rest in the pan for another half minute. Slide the dish directly onto a plate and serve immediately.