Heat butter in a large stockpot over medium-high heat until melted (or heat the olive oil until it is shimmering).  Add onion, carrots, and celery.  Saute for 6-7 minutes, stirring frequently, until the onion is soft and translucent.  Add in the garlic and mushrooms, and saute for an additional 2-3 minutes, stirring frequently, until the garlic is fragrant. Stir in the flour until it is completely combined, and saute for an additional 1 minute, stirring occasionally.  Gradually add in the chicken stock and milk, stirring frequently.  Stir in the chicken, potatoes, corn, peas, Italian seasoning, salt and pepper until combined. Bring the mixture to a simmer (be careful not to bring it to a boil!), stirring occasionally so that the bottom of the soup does not stick to the pan.  Then reduce heat to medium-low, and continue simmering the soup for 10-15 minutes until the potatoes are soft and cooked through. Taste the soup and season with additional salt and pepper, if needed. Serve immediately, with crackers on top or on the side, if desired.