Place lemon zest, garlic and parsley on a cutting board. Gather and chop repeatedly until mixture resembles coarse sand. Set aside.
Place chicken between 2 sheets of waxed paper and pound flat with a rolling pin. Season with salt and pepper.
Melt butter in a large skillet with oil over medium-high heat. Add chicken; cook until browned, 2 to 3 minutes per side. Transfer to a plate and cover loosely with foil to keep warm.
Pour off fat from skillet and add broth. Increase heat to high and bring to a boil, stirring to pick up browned bits. Boil until liquid has thickened and reduced to 2 Tbsp., about 2 minutes.
Return chicken to skillet; top with cheese. Reduce heat to medium and cover. Heat until cheese is melted and chicken is cooked through, 3 minutes. Transfer to plates and drizzle with juices from skillet. Sprinkle with lemon-garlic mixture.