Beat eggs and milk together in a shallow bowl. Whisk flour, salt, black pepper, and cayenne pepper together in another shallow bowl.
Gently press chicken into flour mixture to coat; shake off excess flour. Dip into egg mixture and coat completely.
Heat olive oil and 1 tablespoon butter together in a large skillet over medium heat. Cook chicken in hot oil until lightly browned, 2 to 3 minutes per side. Transfer chicken to a plate.
Cook and stir vegetable broth, sherry, and lemon juice in the same skillet; bring to a boil and cook until sauce is reduced by half. Remove from heat and add cold butter chunks; stir until butter is melted and sauce is shiny and thickened, 1 to 2 minutes. Stir parsley into sauce and season with salt and pepper.
Return chicken to the skillet over medium heat, spoon sauce over the top, and cook until chicken is heated through and no longer pink in the center, 1 to 3 minutes more.
