1. Combine 1 teaspoon thyme, 1/4 teaspoon salt and 1/8 teaspoon pepper in a cup. Sprinkle over both sides of each chicken breast. Heat 1 tablespoon oil in a large skillet over medium-high heat. Add chicken to skillet and cook until golden and cooked through, turning once, about 7 minutes. Transfer to a platter and cover.
2. Cook pasta according to package directions. Meanwhile, heat 1 tablespoon oil same skillet over medium-high heat. Stir in mushrooms and onion and cook, stirring occasionally until onion is translucent, about 7 minutes. Stir in garlic and cook 1 minute more. Transfer to a large bowl. Whisk together broth, flour, and mustard, then add to skillet and bring to a simmer, about 1 minute. Stir in tomatoes, parsley, and remaining thyme and cook 1 minute more.
3. Transfer drained pasta to a large serving bowl and toss with remaining oil, salt, and pepper, then stir in mushroom mixture and hot broth mixture. Serve with chicken.