In a saucepan, bring broth and lemon juice to a boil. Stir in rice and onion; return to a boil. Reduce heat; simmer, covered, until liquid is almost absorbed and rice is tender, 15-20 minutes.
Meanwhile, in a shallow bowl, whisk together egg and milk. In another bowl, toss bread crumbs with 1/3 cup cheese and oregano. Pound chicken breasts with a meat mallet to 1/4-in. thickness. Dip in egg mixture, then in crumb mixture to coat both sides.
In a large skillet, heat oil over medium heat. Cook chicken until golden brown and chicken is no longer pink, 2-3 minutes per side.
When rice is cooked, gently stir in peas; cook, covered, until heated through, 1-2 minutes. Stir in lemon zest and remaining cheese. Cut chicken into slices; serve with rice. If desired, sprinkle with pepper.