Bring stock to a boil in a stockpot. Heat olive oil in a skillet over medium heat. Add carrot and onion; cook and stir until onion is translucent, 5 to 7 minutes. Scrape carrot and onion into to stock.
Stir potatoes and bay leaf into stock; cook and stir until potatoes begin to soften, 10 to 15 minutes. Stir amaranth leaves and garlic scapes into stock; continue to cook and stir until soup flavors have melded and potatoes can be easily pierced with a fork, 5 to 10 minutes more. Stir scallions, parsley, salt, and pepper into soup.
Divide chicken among 4 serving bowls. Pour soup over chicken to serve.
