Preheat your oven to 400°F (200°C). Cut the tops off the heads of garlic, drizzle with olive oil, sprinkle with salt and pepper, wrap in foil, and roast for 40 minutes.

Once roasted, cool the garlic, then squeeze the cloves into a mixing bowl to create a paste.

In a Dutch oven, heat the butter and olive oil over medium heat. Sauté the shallot for about 3 minutes until translucent.

Incorporate both roasted and smashed raw garlic, cooking for 2 minutes until fragrant.

Pour in chicken broth, add whole cloves, sage, thyme, and bay leaf. Bring to a boil, reduce heat, and simmer uncovered for 15 minutes.

Remove the bay leaf and blend the soup until smooth.

Return the blended soup to low heat, stir in heavy cream, and prepare egg mixture by whisking yolks with vinegar.

Gradually drizzle egg whites into the soup while stirring, then temper the yolks before adding.

Season with salt and pepper, simmer for an additional 5 minutes.

Ladle the soup into bowls, garnish with thyme and Gruyere cheese.