Sprinkle chicken with 1 tsp. each salt and pepper. In a large, heavy frying pan over medium-high heat, melt 1 tbsp. butter with the oil. Add chicken in a single layer and cook, without moving, until browned underneath, about 5 minutes. Turn over and cook until second side is lightly browned, about 2 minutes. Transfer chicken to a plate; drain fat from pan.
Meanwhile, trim fennel bulbs (chop and reserve 1/4 cup fronds) and cut bulbs into quarters; core and thinly slice. Trim ends of endive and quarter heads lengthwise. Chop shallot.
Return pan to medium-high heat. Add wine and broth and stir, scraping bottom, to release browned bits from pan. Add chicken, fennel, endive, shallot, and remaining butter. Cover and cook, stirring occasionally (pan will be very full at first), until chicken is no longer pink at the bone, about 15 minutes.
Add cream to pan, increase heat to high, and cook uncovered, stirring, until slightly thickened, about 5 minutes. Season with salt and pepper. Spoon into a shallow bowl and sprinkle with parsley and reserved fennel fronds.