Bring a large pot of lightly salted water to a rapid boil over medium-high heat; add chicken breasts, cover, and cook until tender, 15 to 20 minutes. Drain and transfer chicken to a cutting board; use two forks to shred chicken and set aside.

Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a 10x15-inch baking dish.

Melt butter in a skillet over medium heat. Cook onion, green bell pepper, and red bell pepper in the melted butter until just soft, 5 to 8 minutes; transfer to a large bowl. Add shredded chicken, Cheddar cheese, green chile peppers, salsa, cilantro, cumin, and chipotle peppers to the bowl. Season with salt and pepper; mix.

Place about 1/3 cup of chicken mixture along the edge of a tortilla; roll tightly into a tube. Place enchilada, seam-side down, into the prepared baking dish. Repeat until all of the chicken mixture is used. Sprinkle Monterey Jack cheese over enchiladas. Whisk whipping cream and chicken broth together in a small bowl; pour over enchiladas. Cover the dish with aluminum foil.

Bake in preheated oven 30 minutes. Remove foil and continue baking until cheese is completely melted, about 10 minutes.