Preheat the oven to 375°F and have ready a greased 9x13 casserole dish.
Add butter to a large skillet or Dutch oven over medium heat and allow to melt. Add the onion and mushrooms to the pan and sauté 5-7 minutes, stirring occasionally.
Sprinkle the flour over the mixture and stir to coat. Add the chicken broth a splash at a time, whisking after each addition, until the mixture resembles pudding. Stream in the half and half. Stir to combine and allow to simmer and slightly thicken about 6 minutes. The sauce will taste salty.
Remove from the heat and stir in half of the Monterey Jack and half of the Cheddar, along with the salt and pepper. Add the shredded chicken and cooked rice and stir to mix. Transfer the contents to the casserole dish.
Top with remaining cheese, cover with aluminum foil, and bake for 15-20 minutes, or until the cheese has melted.