In a large pot, heat the olive oil and butter over medium heat.
Add the chopped onion and sauté until softened, about 5 minutes.
Add the peeled garlic cloves to the pot and cook for 2-3 minutes, stirring frequently to avoid burning.
Pour in the broth and add the thyme and bay leaves.
Bring the soup to a boil, then reduce the heat and let it simmer for 30 minutes until the garlic is soft and fragrant.
Remove the thyme sprigs and bay leaves, and season the soup with salt and pepper.
In a small bowl, whisk the egg yolks and Parmesan cheese together.
Gradually whisk a ladle of hot soup into the egg mixture to temper it, then slowly whisk the tempered mixture back into the soup.
Simmer for another 5 minutes, stirring continuously until the soup thickens slightly.
Serve hot with crusty bread and garnish with fresh parsley if desired.