Put the garlic halves, cut side down, on a sheet of aluminum foil on a rimmed baking sheet. Add 2 tablespoons chicken stock and wrap the foil around the garlic, keeping the cut sides flat on the sheet tray. Roast until the garlic is very soft, about 30 minutes. Remove the garlic from the oven and let stand until cool enough to handle.
Squeeze the soft garlic from the head with your fingers into a large saucepan. Whisk in 1 cup of the stock to loosen the garlic paste and put the pot over medium-high heat. Add the remaining stock, lemon juice, and chopped tarragon; season with salt and pepper, to taste. Bring the mixture to a boil, then reduce the heat to a simmer and add the chicken. Cover and cook until the chicken is just cooked through, about 9 to 12 minutes.
Ladle the soup into bowls and garnish with thin slices of lemon and a sprinkle of fresh tarragon leaves.