In a large sauce pot over medium heat, saute celery and carrots in the olive oil for about 5 minutes. Add in the mushrooms and saute for another 2 minutes.
Add the chicken broth, water, and chicken to the sauce pot. Bring to boiling, and then stir in rice (saving the seasoning packet for later). Cover and turn heat to low.
In a medium saucepan over medium heat, melt butter and then stir in seasoning packet until the mixture starts to bubble. Reduce heat to low and slowly whisk in flour, tablespoon by tablespoon, until it makes a thick paste. Slowly whisk in half and half until completely smooth. Continue cooking and stirring until it starts to thicken (about 5 minutes).
Stir cream mixture into the broth mixture and cook until heated through (about 10 minutes).
Notes
Fridge: For any leftover chicken and rice soup place it in a large airtight container to serve the next day or up to 4 days. Freezer: If you are like me and want to double the batch to store more for the soup season ahead for those cold days, let the soup get to room temperature and pour it into a freezer bag and you can freeze cream-based soups for up to 60 days.