Season the chicken with ½ teaspoon of the salt and ¼ teaspoon of the black pepper.
In a large skillet, heat 1 teaspoon of the olive oil over medium-high heat. Add the chicken breasts and cook until the chicken is no loner pink in the center but still juicy, 4 to 6 minutes per side. Set aside on a plate to rest.
Meanwhile, bring a large pot of salted water to a boil over high heat. Cook the pasta to al dente according to package directions. Reserving ½ cup of the pasta water, drain and set aside.
Add the remaining 1 teaspoon of oil to the same skillet used for the chicken, and sauté the broccoli, bell peppers, and mushrooms for 5 to 7 minutes, until the vegetables soften. Season with the remaining ½ teaspoon salt and ¼ teaspoon pepper. Remove the vegetables from the skillet and set aside.
To make the Alfredo sauce: In a medium saucepan, melt the butter over low heat. Whisk the flour into the butter for about 30 seconds, until smooth and the mixture smells nutty, making a roux. Whisk in the milk and bring to a boil over medium heat, stirring constantly.
Reduce the heat to low and stir in the cream cheese and Parmesan. Stir the Alfredo sauce frequently, until it is slightly thickened, 3 to 4 minutes.
Slice the chicken diagonally in thin strips. Return the chicken and vegetables to the skillet and toss to combine.
Transfer the pasta to the chicken and vegetables to the skillet. Coat the chicken and vegetables evenly with the Alfredo sauce and add the reserved pasta water as needed until the sauce reaches a desired consistency before serving. Serve warm.