Prep the Veggies and Aromatics: Heat olive oil and butter over medium heat until the butter melts and foams. Add diced onions, carrots, and celery with a pinch of salt, stirring occasionally until they soften, about 7–10 minutes. This builds your flavor base and brings out their natural sweetness.
Build the Roux and Sauce: Stir in garlic and Italian seasoning, cooking until fragrant—about 30 seconds. Sprinkle in the flour and whisk constantly for a minute to remove the raw taste. Slowly pour in chicken broth while whisking, scraping any browned bits off the bottom. Pour in milk and simmer until slightly thickened, about 3 minutes.
Add Chicken and Peas: Stir in the shredded rotisserie chicken and frozen peas, then simmer gently for another 10 minutes so the flavors meld and veggies soften. The sauce should coat the spoon nicely; if it feels thin, simmer a bit longer or thicken with a slurry.
Finish and Serve: Remove from heat, stir in fresh parsley, and season with salt and pepper to taste. Serve hot—either as the filling of a pot pie, over biscuits, or with a side salad. It’s rich and comforting all on its own.