Sauté the Aromatics: Heat the olive oil and butter in a large pot over medium heat. Add the chopped onion and cook until soft and translucent, about 5 minutes. Add the peeled garlic cloves and continue cooking for another 2-3 minutes until fragrant but not browned.
Simmer the Soup Base: Pour in the chicken or vegetable broth and add the fresh thyme sprigs and bay leaves. Bring the mixture to a boil, then reduce heat and simmer uncovered for 30 minutes to allow the flavors to meld together. Remove the thyme sprigs and bay leaves.
Season the Soup: Add salt and pepper to taste, adjusting the seasoning according to your preference.
Enrich the Soup: In a separate bowl, whisk together the egg yolks and grated Parmesan cheese until well combined. Gradually add a small amount of the hot soup to the egg mixture, whisking constantly to temper the eggs (this prevents them from curdling). Slowly pour this mixture back into the pot with the remaining soup, whisking continuously.
Finish and Serve: Let the soup simmer for about 5 minutes until slightly thickened, being careful not to let it boil. Ladle the soup into bowls and garnish with fresh chopped parsley before serving.