Dice the onion, celery, carrots, and green bell pepper into small ¼-inch (6mm) pieces, peel and cube the russet potato into ½-inch pieces, mince the garlic, and roughly chop the fresh parsley. Uniform sizing ensures even cooking throughout.
Heat olive oil in a large Dutch oven over medium heat until shimmering, then add the onion, celery, carrots, and green bell pepper and sauté for 5 to 6 minutes until softened. Add minced garlic, dried oregano, and dried basil, stirring constantly for 1 minute until fragrant.
Add the diced potato, canned diced tomatoes with their juices, chicken broth, and water to the pot. Stir in the chopped parsley, kosher salt, black pepper, white pepper, and red pepper flakes until fully combined.
Nestle the bone-in chicken thighs into the liquid and bring to a boil over medium-high heat, then reduce to a gentle simmer. Partially cover and cook for 45 minutes to 1 hour until the chicken is fall-apart tender and the broth has deepened in flavor.
Use tongs to transfer the chicken thighs to a cutting board and let them cool for 5 minutes. Discard the skin and bones, then shred the meat into bite-sized pieces using two forks or meat shredding claws and set aside.
Use a potato masher to lightly press on some of the softened potatoes and vegetables directly in the pot to slightly thicken the broth without fully pureeing it. Stir to incorporate the mashed vegetables evenly throughout.
Bring a separate saucepan of salted water to a boil and cook the ditalini pasta 1 to 2 minutes less than the package directions indicate (al dente). Drain through a colander and set aside.
Return the shredded chicken to the pot, add the drained ditalini, and stir gently to combine. Simmer together for 5 minutes, then taste and adjust salt, pepper, and red pepper flakes as needed before ladling into bowls.
Ladle the hot soup into bowls and top with freshly grated Parmesan cheese if desired. Serve immediately with crusty bread or warm dinner rolls on the side.