Start by chopping your onion, carrot, and celery. If your chicken is not already cooked, cook and shred it now. Thaw your peas if using frozen ones. Get all your ingredients ready before starting—this will make cooking easier and faster.
In a large pot, melt butter over medium heat. Add the diced onion, carrots, and celery. Stir them gently until they soften and begin to smell aromatic, usually around 5–6 minutes. These veggies form the base flavor for that "pot pie" taste.
Sprinkle in the all-purpose flour and stir well. Let it cook for a minute so the flour doesn’t taste raw. Slowly pour in the chicken broth while whisking. This dissolves the flour mixture and gives you a lump-free base.
Add in the whole milk and bring everything to a light simmer. Stir often to prevent the milk from scalding. Then add the dry egg noodles and let them cook right in the soup—this infuses more flavor into them.
Once the noodles are almost done, toss in the shredded chicken and peas. Let it all cook together for another 3 minutes or so until warmed through. Season with salt and pepper to taste.