Preheat your oven to 400°F (200°C). Cut the top off one whole head of garlic to expose the cloves. Drizzle it with a bit of olive oil, sprinkle with sea salt and ground black pepper, and wrap it completely in foil. Place the wrapped garlic on a shallow baking dish and bake for 40 minutes. Allow it to cool slightly, then squeeze the roasted garlic cloves out of the head—they should pop out easily.

Heat the unsalted butter and the remaining olive oil over medium heat in a Dutch oven. Slice the shallot and add it to the pot, softening for about 3 minutes, until translucent. Smash all the roasted garlic cloves from Step 1 along with the remaining raw garlic cloves from the second head. Add both roasted and raw garlic to the pot and cook, stirring, for another 2 minutes until fragrant.

Pour in the chicken broth and add the whole cloves, chopped fresh sage, dried thyme, and bay leaf. Bring the mixture to a gentle boil and cook for 5 minutes. Then lower the heat to a simmer and cook for 10 more minutes to blend the flavors. Discard the bay leaf. Carefully blend the soup until smooth using an immersion blender or work in batches with a standard blender—just be sure to let the soup cool slightly before blending to avoid splatters.

Return the blended soup puree to your Dutch oven over a low simmer. Stir in the heavy cream until fully incorporated. In a small bowl, whisk together the egg yolks and red wine vinegar. In a separate small bowl, lightly whisk the egg whites. This will prepare the eggs for enrichment. I like to use heavy cream here for a luscious, smooth texture.

Gradually drizzle in the whisked egg whites into the hot soup, whisking to create delicate, stringy ribbons throughout. Next, temper the yolks by slowly adding a bit of hot soup to the yolk mixture while stirring, then drizzle the tempered yolks into the pot, stirring constantly. Add the remaining sea salt and ground black pepper, increase the heat to medium, and cook the soup for another 5 minutes until slightly thickened. For a richer flavor, I like to add a bit of extra black pepper at the end.

Ladle the finished soup into cups or bowls. Garnish each serving with a sprinkle of dried thyme leaves and grated Gruyere cheese. Serve immediately while hot.