Rinse barley. Chop onion. Stem chard by cutting away leafy green prtion from either side of stalk. Set leaves aside. Chop 1 cup chard stalks and mince garlic. Heat two tablespoons oil in large saucepan over medium high heat until shimmering. Add onion and chard stalks and cook until softened, about five minutes. Stir in barley and garlic and cook until barley is lightly toasted, about three minutes.
Stir broth into saucepan with barley and bring to simmer. Reduce heat to low, cover, and simmer until barley is tender and broth is absorbed, 20 to 40 minutes.
While barley cooks, cut chard leaves into 1 inch pieces. Grate 1 teaspoon lemon zest, then cut lemon in half and remove seeds. Grate parmesan (1/2 cup).
Fold chard leaves and lemon zest into barley, increase heat to medium high and cook uncovered , stirring gently, until chard is wilted about two minutes. Off heat, stir in parmesan cheese and season with salt and pepper. Transfer to serving platter and tent with aluminum ofil.
Trim cutlets, pat dry and season with salt and pepper. Heat two tablespoonsoil in 12 inch non stick skillet over medium high heat until shimmering. Add lemon halves cut side down andcook until browned, about two minutes, transfer to platter with barley.
Heat two teaspoons oil in now empty skillet and heat until shimmering. Add 3 cutlets to skillet and cook until well browned and tender about 2 minutes per side. Add to platter. Repeat with oil and the 3 additional cutlets. Squeeze lemon halves over cutlets. sprinkle with remaining parmesan cheese.