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. These can be bitter, especially when you are using lots of garlic in a recipe, so it's best to remove them. Use the tip of a sharp knife or a toothpick to pry them out 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. Take a few minutes to get it all finely mashed. Add the mashed garlic to the stock and continue to gently simmer for 10 minutes.
Now make the mayonnaise. Put the egg yolk, mustard, salt and oil in a small jar, just wide enough to fit your immersion blender. Note: reserve the egg white, you may need it later. Place the blender head at the bottom of the jar and blend, gently lifting the blender slightly to get all of the contents emulsified. Note: if your mayonnaise does not instantly become thick and creamy, add the egg white and blend again.
I like to strain my soup at this point, because I want a velvety smooth texture, but this is optional. If you do strain, put the strained soup back in the pot.
Take a ladle-ful of the hot soup and stir it into your mayonnaise to temper it. Then immediately stir the mayonnaise into the soup pot, whisking as you add it.
Bring the soup back on the stove and bring up to temperature, but do not let it boil or you risk curdling it.
Serve with crusty croutons, an extra drizzle of olive oil, a sprinkle of smoked paprika, and fresh parsley if you like.