How to Make It
Thinly slice 2 heads of garlic (or, for speed, pulse them in a food processor until roughly chopped). Heat 2 tablespoons of the oil in a large pot over medium-low heat. Add the garlic, sage, 1 teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until soft and fragrant, 6 to 8 minutes.
Add the chicken thighs and 8 cups water. Bring to a boil, reduce heat, and gently simmer, skimming off any foam from the surface, until the chicken is cooked through, 12 to 15 minutes. Transfer the chicken to a plate and let cool slightly before shredding the meat. Return the meat to the broth.
In a large, deep skillet, add the vinegar to 3 inches of barely simmering water. Crack the eggs into a teacup, then slip them into the water. Cook until the whites are set, 2 to 3 minutes. Remove with a slotted spoon.
Drizzle the toasted baguette with the remaining oil and rub with the remaining garlic.
Serve the soup topped with the eggs, toasts, scallions, and Parmesan.