Sprinkle 4 chicken cutlets with ½ teaspoon salt and ¼ teaspoon pepper.
Heat 1 tablespoon oil in a large skillet over medium heat.
Add the chicken; cook, turning once, until browned and an instant-read thermometer inserted in the thickest part registers 165°F, 6 to 8 minutes.
Transfer to a platter and loosely cover with foil to keep warm.
Add 4 cups broccoli and the remaining 1 tablespoon oil to the skillet.
Increase heat to medium-high; cook, stirring occasionally, until the broccoli is lightly seared and bright green, about 1 minute.
Stir in grated garlic; cook, stirring constantly, until fragrant, about 30 seconds.
Add ½ cup wine; cook until slightly reduced, about 30 seconds.
Reduce heat to medium and stir in ½ cup cream and the remaining ¼ teaspoon each salt and pepper.
Simmer until the broccoli is tender-crisp and the sauce is slightly thickened, 3 to 4 minutes.
Spoon the broccoli and sauce over the chicken.
Garnish with parsley, if desired.
