For the soup, in a large pot, add the broth and bring to a simmer. Stir in the frozen egg noodles and simmer for 10 minutes.
Add the diced carrots, celery, onions, bay leaf, parsley, and salt. Cover the pot, fully or partially, and simmer for 10 to 15 minutes, until the noodles and veggies are tender, stirring now and then to make sure the noodles aren't sticking.
For the roux, while the soup simmers, in a small saucepan, add the olive oil and butter and cook on medium heat until the butter is melted. Add the flour and whisk to combine until a smooth paste forms. Remove from the heat.
Add the milk to the soup and heat through. While stirring, scrape the roux into the soup and cook, stirring or whisking constantly, over medium heat until the soup has thickened slightly, 5 to 7 minutes.
Stir in the chicken, season to taste with salt and pepper (this is important for flavor!) and heat through.
Remove the bay leaf. Serve immediately or keep warm until ready to serve. The soup will thicken more as it cools.
Notes
Egg Noodles: look for frozen egg noodles (usually near the bread products in the freezer section). They are usually cut into straight, thick pieces. Two popular brands are Grandma's and Reames. I don't thaw the frozen noodles before using. They are clumped together in the package, but once they cook for a few minutes in the hot broth, they separate nicely.If you can't find frozen egg noodles, use dried old-fashioned or "country" style egg noodles (they are thicker and heartier than regular dried egg noodles). If using dried noodles, you may need to adjust the cooking time for the noodles in the recipe.