Place chicken and garlic in a soup pot. Add chicken broth and bring to boil; reduce heat to low and simmer, covered, until chicken is no longer pink in the center and juices run clear, about 30 minutes. Remove from pot; let cool 10 minutes. Dice into bite-sized pieces. Set aside.

Bring a large pot of lightly salted water to a boil. Cook tortellini in the boiling water, stirring occasionally, until tender yet firm to the bite, 10 to 12 minutes. Drain and set aside.

Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch oven-proof glass container (such as Pyrex®) with cooking spray.

Melt butter in a saucepan over medium heat. Add green onions, 1 cup Monterey Jack cheese, sour cream, cream of mushroom soup, and cream of chicken soup; cook until heated through, 5 to 7 minutes.

Combine diced chicken, tortellini, cream sauce, and button mushrooms in a large bowl. Pour into the prepared 9x13-inch Pyrex® baking dish. Top with remaining cup of Monterey Jack cheese.

Bake in the preheated oven until bubbling and top is golden brown, 30 to 45 minutes.