In a large deep pot over medium-low heat melt the butter. Once butter has melted add in the chopped onions, celery, carrots, 1 tsp of salt and a few cracks of fresh pepper. Cook for 5-7 minutes until onions have sweated out are soft and translucent.
Mix in garlic and thyme. Cook for another 2 minutes then add the flour. Toss the par cooked vegetables in the flour and cook for 5 minutes stirring every few seconds to ensure that the flour does not burn at the bottom of the pot.
Add both chicken and beef broth to the pot and mix vigorously to incorporate the flour into the broth. Add the chicken and peas. Bring to a boil with the lid on over medium heat (approx 5-7 minutes). Once at a boil add in the heavy cream and bring to a soft boil with the lid off for 20 minutes. Soup will still be thin by the end of the 20 minutes.
Add the grated parmesan and wild rice. Cook for another 10 minutes.
Remove off the heat and let the soup sit for 5-10 minutes before serving, during this time the soup will cool down, and thicken up.