In a large pot, heat olive oil and butter over medium heat until melted and foamy. Sauté chopped onion for about 5 minutes until soft and translucent.

Stir in peeled garlic cloves, cooking for 2-3 minutes until fragrant and lightly golden.

Pour in chicken or vegetable broth, and add fresh thyme and bay leaves. Bring to a boil, then reduce to a simmer for 30 minutes.

Remove thyme sprigs and bay leaves. Season with salt and pepper to taste.

Whisk together egg yolks and grated Parmesan in a bowl. Temper with a ladle of hot soup, then whisk back into the pot and simmer for another 5 minutes.

Ladle the hot soup into bowls and serve immediately with crusty bread and optional parsley.