Heat one tablespoon of olive oil over medium-high heat in a large pot or dutch oven. Add the chicken thighs and brown on both sides, about 4-5 minutes each. You might have to cook the chicken in a couple of batches because you don't want to overcrowd the pot.
Remove chicken and set aside.
In the same soup pot, pour in the rest of the olive oil and add the carrots, celery, onion, garlic. Cook over medium heat for about 5 minutes or until fragrant and softened a bit.
While the vegetables are cooking, chop the chicken thighs into bite-size pieces.Set the chicken aside.
Pour the wine to the soup pot. As the wine boils, scrape up any bits that got stuck to the bottom of the pan. These bits are little nuggets of flavor, so don't skip this step. If you don't have wine, then use chicken broth.
Stir in the chopped chicken and chicken stock. Bring to a boil, reduce heat and let simmer for about 15 minutes.
Slow stir in the milk and rice. Bring to a slow boil and simmer for another 5 minutes or until hot.
Add in the salt and black pepper. Give it a taste and add more salt and pepper if needed.
Serve in small bowls with grated romano cheese.