In a stockpot or large pot, add the chicken, bones (if using), carrots, onion, garlic, peppercorns, bay leaf, and a pinch of salt. Add just enough water to cover the chicken, about 11 cups (add a little more only if needed). Bring to a boil over high heat, then lower the heat to maintain a moderate simmer. Cook, skimming any impurities that rise to the surface, until the broth is richly flavored and the carrots are fully tender, about 1 hour and 10 minutes. Set a fine-mesh strainer over another large pot or a large heatproof bowl that can fit at least 10 cups. Strain the broth. Let the solids rest in the strainer until they’re cool enough to handle. Meanwhile, make the rice: In a small pot, add rice, 1½ cups water, and a generous pinch of salt. Bring to a strong simmer over medium-high heat, then reduce the heat to maintain a very low simmer and cover the pot. Cook until the water is fully absorbed and the grains are tender, about 15 minutes for white rice and 20 minutes for brown. Turn off the heat and let the rice rest, covered. Using your fingers, pull the chicken meat from the bones, prioritizing the meat you like best (you may have some left over) and shredding it into small strands. Transfer the carrots to a medium bowl and smash into bite-size chunks with the back of a fork (you will have about 2 1/4 cups). Add the carrots and 2 1/4–2/12 cups of chicken back to the pot with the broth. (Reserve any remaining chicken for another use.) Retrieve the garlic and squeeze the cloves into the broth, if desired. Stir in the tomato paste and fish sauce until incorporated. When ready to serve, bring the soup back up to a gentle simmer. Gently fluff the rice with a fork, then stir it into the broth. Add the kale and cook until the rice is warmed through and the kale is just wilted but still bright green, about 3 minutes. Taste and adjust the seasoning with salt as needed. Ladle the soup into bowls. Top generously with pepper, Pecorino Romano, and red chile flakes to taste.