Method:
Put dried mushrooms into a heatproof bowl with 1/2 cup boiling water; set aside to let soak. Meanwhile, cook pasta in boiling salted water according to package directions until al dente; drain well.
Warm 1 tablespoon of the oil in a large skillet over medium heat. Add chicken and cook, stirring frequently, until lightly browned, about 4 minutes. Transfer with a slotted spoon to a plate and set aside.
Add remaining 1 tablespoon olive oil and button mushrooms to the same skillet and cook, stirring frequently, about 3 minutes. Add bell peppers and cook 3 minutes longer. Stir in broth, sage, salt and pepper.
Drain soaked mushrooms and transfer soaking liquid to skillet. Coarsely chop soaked mushrooms and add to the skillet; bring to a boil. Add broccoli and chicken, reduce heat to a simmer, cover and cook until broccoli is warmed through and chicken is cooked, 3 to 5 minutes.
Transfer sauce to a large serving bowl. Toss with cooked pasta and crème fraîche and serve immediately.