Sauté the aromatics. Heat a drizzle of oil and the butter in a large pot over medium heat.
Add onion, celery, and carrots with a pinch of salt. Cook 5–7 minutes, stirring, until softened.
Add garlic and herbs. Stir in garlic, thyme, and the bay leaf. Cook 30–60 seconds until fragrant.
Make a light roux. Sprinkle the flour over the veggies and stir for 1–2 minutes.
It should look a bit pasty and coat the vegetables.
Whisk in broth. Slowly pour in the chicken broth, whisking to avoid lumps. Add the diced potatoes. Bring to a gentle simmer and cook for 10–12 minutes, until the potatoes are tender.
Stir in milk. Lower the heat and add the milk.
Simmer another 3–5 minutes to thicken slightly. Avoid boiling to keep the milk from curdling.
Add chicken and veggies. Stir in the cooked chicken, peas, and corn. Simmer 3–4 minutes, just until heated through.
Finish and season. Remove the bay leaf.
Add parsley and a squeeze of lemon. Season with salt and pepper to taste. If you like it thicker, let it simmer a few more minutes; for thinner, add a splash of broth.
Serve. Ladle into bowls and garnish with extra parsley or cracked pepper.
For a pot-pie touch, serve with whole-grain crackers or a piece of warm, crusty bread.