Heat the oil in a medium-sized soup pot with a thick base, over medium-high heat. Add the onion, mushrooms, salt and pepper, and cook for 2 minutes, mixing with a wooden spoon.
Add the carrots, celery, butternut squash and zucchini, garlic, tomato, baby spinach and parsley, mix and continue cooking for 3 minutes.
Cover with 4 cups (1 L) of water and boil. Then lower the heat to medium and cook for 8 minutes without the lid.
Lower the heat a little to medium-low. Crack eggs on top of the soup, cover and cook for 3 minutes. Turn off the heat (remove the pot of soup from the stove) and let stand for 3 minutes for a softer egg yolk or 5 minutes for a firm yolk.
To serve: In each bowl, serve the soup with the eggs and add a tablespoon (15 mL) of your cheese to the top. You can also garnish the dish with Espelette pepper or chilli flakes and serve with a slice of toasted bread!