Heat the olive oil in a large pot over medium heat. Add the chopped onion and cook until soft and translucent, about 5 minutes.

Add the garlic cloves and cook gently for 2–3 minutes, stirring often, until fragrant but not browned.

Pour in the broth and add the bay leaf and thyme. Bring to a boil, then reduce heat and simmer uncovered for 20–25 minutes, until the garlic is very soft.

Remove the bay leaf and thyme. Blend the soup until smooth using an immersion blender or by carefully transferring it to a blender.

In a small bowl, whisk together the egg yolks and grated cheese.

Slowly ladle a small amount of hot soup into the egg mixture while whisking constantly to temper it.

Gradually stir the tempered egg mixture back into the pot.

Heat gently over low heat for a few minutes, stirring constantly, until the soup thickens slightly. Do not boil.

Season with salt and pepper to taste.

Serve warm, garnished with parsley and toasted bread cubes if desired.