Heat the olive oil in a large pot over medium heat.
Add the diced onion and sauté until soft, about 5 minutes.
Stir in the minced garlic and cook for an additional 1-2 minutes until fragrant.
Add the chopped Swiss chard and kale, cooking until wilted, about 5 minutes.
Pour in the vegetable broth and season with salt and pepper.
Bring the soup to a gentle simmer and cook for 15 minutes.
Stir in the chopped spinach and fresh thyme, cooking for another 5 minutes.
Adjust consistency with water if needed, then taste and adjust seasoning.
Serve hot, garnished with a drizzle of olive oil or a sprinkle of fresh herbs if desired.