Pour the chicken stock into a large Dutch oven or stockpot and add the fresh sage. Set over medium heat and bring to a gentle simmer, then reduce the heat to low and keep it warm while you prepare the garlic.

Peel the garlic cloves, then slice each one in half lengthwise and remove the tiny green germ in the center if present.

Bring a small saucepan of water to a rolling boil, add the garlic halves, and cook for 3 to 4 minutes. Drain well.

Using the back of a fork, mash the blanched garlic until smooth. Stir the mashed garlic into the simmering stock and let it cook gently for 10 minutes.

In a small jar, combine the egg yolk, Dijon mustard, salt, and olive oil. Insert an immersion blender to the bottom, then blend without lifting until the mixture emulsifies into a thick mayonnaise.

If the mayonnaise seems too thin, whisk in the reserved egg white a little at a time until it reaches a spoonable consistency.

For the silkiest texture, strain the soup through a fine-mesh sieve into a clean pot, discarding the sage and garlic solids. You can skip this step for a more rustic soup.

Ladle a cup of the hot soup into the mayonnaise and whisk to temper it, then pour the warmed mayonnaise back into the pot while whisking constantly.

Return the soup to very low heat and warm it through gently. Do not let it boil, or the mayonnaise may curdle.

Ladle the soup into bowls and finish with crusty croutons, a drizzle of olive oil, a sprinkle of smoked paprika, and fresh parsley leaves.