Place a pot of water on the stove and bring it to a boil. Add about a tablespoon of salt to the water and then cook the pasta until it is tender. Towards the end of the cooking time, scoop out a bit of the cooking water and set it aside.
Just before the noodles are done cooking, add the broccoli to the pot. Let the broccoli boil for about 30 seconds, only until it is bright green, and then drain everything into a large colander.
While the pasta is cooking, heat the oil in a large pot over medium-high heat. Add the onions and saute for about 4-5 minutes, until they are lightly browned.
Add the bell peppers and mushrooms and stir to combine. Sprinkle with ½ teaspoon salt and ¼ teaspoon of pepper. Saute for 2-3 minutes until they are hot and slightly tender, but still a bit crisp. Add the garlic and saute until fragrant, about 30 seconds. Stir in the chicken.
Transfer the vegetable mixture to a large mixing bowl and set aside next to the stove. Pour the cream into the empty pot over medium heat. Add the chipotle peppers, 1 teaspoon salt, ¼ teaspoon pepper, oregano, and cayenne. Whisk to combine and bring to a low simmer for about a minute.
Add the pasta to the pot with the cream and toss to coat. Add the vegetables and chicken to the pasta and toss again to mix throughout. Add a splash or two of the pasta water to the pot, if the pasta seems dry. (If you forgot to reserve the pasta water, use a splash of chicken broth.) Serve warm.