Pound out chicken breasts so they are of equal thickness.
Heat oil in a large skillet over medium. Season chicken with salt and pepper, then place into hot skillet.
Cook 3 minutes on each side or until no longer pink; transfer to a plate. While chicken cooks, blanch broccoli; cut into smaller pieces, then set aside.
Chop shallot and saute in same skillet for 1 minute. Add flour and cook for 1 minute more, stirring continuously. Slowly add white wine, continuing to stir. Scrape the inside of the pan to remove the brown bits from cooking the chicken and bring to a boil.
Remove from heat; stir in sour cream, mustard, and tarragon. Add broccoli and noodles, then toss to coat and heat if necessary.
Serve pasta topped with chicken and serve apple slices on the side.