Season the chicken cutlets with ½ teaspoon salt and ¼ teaspoon pepper.
Heat 1 tablespoon olive oil in a large skillet over medium heat. Add the chicken and cook for 6–8 minutes, turning once, until golden and fully cooked (165°F internal temperature). Transfer to a plate and cover loosely.
In the same skillet, add broccoli and remaining olive oil. Increase heat to medium-high and cook for about 1 minute until bright green and lightly seared.
Stir in the garlic and cook for 30 seconds until fragrant.
Add chicken broth and apple cider vinegar; cook briefly until slightly reduced.
Reduce heat to medium and stir in the cream, remaining salt, and pepper. Simmer for 3–4 minutes until the sauce thickens and broccoli is tender-crisp.
Return the chicken or spoon the broccoli and sauce over it. Garnish with parsley and serve.