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.