Place the 4 chicken breasts in a large pot and cover with water. Bring to a boil, then simmer until the chicken is tender and cooked through, about 30 minutes. Drain and let cool slightly.
While the chicken cooks, melt 2 tablespoons butter in a large skillet over medium heat. Add the diced onion, diced bell pepper, and sliced portabello mushrooms and sauté until the vegetables are tender, about 5–7 minutes.
Chop or shred the cooked chicken into bite-sized pieces and return to the pot or a large bowl.
Add the sautéed vegetable mixture, 1½ cups cream of chicken soup, and the drained can of tomatoes with peppers to the chicken. Stir until evenly combined.
Spoon about 1 cup of the chicken mixture into the center of each tortilla. Sprinkle some of the shredded Mexican cheese inside each, then wrap each tortilla into a burrito and place seam-side down in a baking dish. Repeat for all 8 tortillas.
Preheat the oven to 350°F (175°C). Bake the burritos for 30 minutes, then remove the dish from the oven, top the burritos with the remaining cheese, and return to the oven for 5 more minutes until the cheese is melted. Total bake time is 35 minutes.
Remove the baking dish and wrap each burrito tightly in foil if desired for storage or transport.
