Season the chicken cubes generously with poultry seasoning or rotisserie chicken seasoning, adding salt to taste depending on whether your seasoning blend already contains salt.
Heat the olive oil and 1 tbsp of the butter in a large braiser or Dutch oven over medium-high heat. Add the chicken in a single layer. Cook, turning occasionally, until fully cooked through and deeply golden on multiple sides, about 6–8 minutes total. Transfer the chicken to a plate.
Once cool enough to handle, shred or break the chicken into smaller bite-sized pieces. Set aside.
Lower the heat to medium and add the remaining 1 tbsp butter to the same pan. Add the onion, carrots, and celery with a pinch of salt. Cook, stirring often, until very soft and fragrant but not browned, about 7–9 minutes, scraping up the browned bits.
Stir in the garlic and fresh thyme and cook for 30 seconds. Sprinkle the flour over the vegetables and stir constantly for about 1 minute.
Slowly pour in the chicken stock and water, stirring well. Bring to a gentle boil and simmer for 30–60 seconds.
Stir in the pastina. Reduce the heat and simmer, stirring frequently, until tender and creamy, about 8–10 minutes.
Stir the shredded chicken back into the pot along with whole milk, starting with ⅓ cup. Add more milk as needed until creamy and spoonable. Stir in peas if using. Taste and adjust seasoning to taste.
Serve hot with crushed Ritz crackers on top or whole crackers on the side.