Place the chicken breasts in a large pot and cover with water. Bring to a boil, then reduce heat and simmer until the chicken is tender, about 30 minutes. Remove chicken and let cool slightly.
While the chicken cools, melt the butter in a skillet over medium heat. Add the diced onion, diced bell pepper, and sliced portabello mushrooms and sauté until softened, about 6–8 minutes.
Chop or shred the cooked chicken into bite-sized pieces and place in a large mixing bowl.
Add the sautéed vegetables, cream of chicken soup, and drained diced tomatoes with peppers to the bowl with the chicken and stir until evenly combined.
Spoon about 1 cup of the chicken mixture into the center of each tortilla. Roll each tortilla into a burrito and place seam-side down in a 9x13 baking dish.
Bake in a preheated 350°F (175°C) oven for 30 minutes. Remove from the oven, sprinkle the burritos with the shredded Mexican cheese, then return to the oven for 5 more minutes until the cheese is melted.
Remove from the oven and wrap each burrito tightly in foil if desired for storage or transport.
If baking from frozen, thaw in the refrigerator overnight and then bake as instructed; if still frozen, bake at 350°F (175°C) for about 1 hour.