Heat the chicken stock and sage in a large Dutch oven or stock pot. Bring to a simmer and then simmer very gently while you prep your garlic.
Remove the skins from the garlic cloves and slice them in half lengthwise. Remove the little embryonic shoot and discard.
Bring a small saucepan of water to a boil and add the garlic cloves. Boil for 3-4 minutes. Drain the cloves.
Use the back of a fork to mash the softened garlic until finely mashed. Add the mashed garlic to the stock and continue to gently simmer for 10 minutes.
Make the mayonnaise by putting the egg yolk, mustard, salt and oil in a jar, and blend with an immersion blender.
If your mayonnaise is not thick, add the reserved egg white and blend again.
Optionally strain the soup for a velvety smooth texture and return it to the pot.
Take a ladle-ful of the hot soup and stir it into your mayonnaise to temper it, then stir the mayonnaise into the soup pot.
Bring the soup back on the stove to heat, but do not let it boil.
Serve with crusty croutons, a drizzle of olive oil, smoked paprika, and fresh parsley if desired.
Notes
Recipe lightly adapted from The French Cooking Academy.