Bring a large pot of salted water to a boil over high heat.
Cook the pasta for the time on the package, about 11 minutes, adding the broccoli for the last 3 minutes. Drain.
If the chicken breasts are thick, slice them horizontally into thin cutlets about 1.5 cm (1/2 inch). Season with the salt and dust lightly with the flour.
Heat the oil in a large skillet over medium-high heat. Sear the chicken for about 4 minutes per side, until golden and 74°C (165°F) inside. Move to a plate.
Add the sliced mushrooms to the pan and cook for 5 minutes until browned.
Add the broth, balsamic vinegar and thyme, scraping up the browned bits, and simmer for 4 minutes until slightly reduced. Swirl in the butter.
Return the chicken to the sauce to warm through and serve with the pasta and broccoli.