Bring enough salted water to a boil in pot to cover egg by 2 inches. Gently lower egg into water and reduce heat to vigorous simmer. Cook 13 minutes. Drain and run cold water over egg until cooled; peel and coarsely chop.
Cook broccoli in boiling salted water in medium pot 4 to 6 minutes or until crisp-tender; drain and reserve.
Meanwhile, melt butter in medium skillet over medium-high heat. Add shallots and cook 1 to 2 minutes until butter and shallots begin to brown. Add bread crumbs, stirring until browned, about 1 minute. Remove pan from stove; stir in parsley, salt and pepper to blend.
Sprinkle crumb mixture over broccoli, topping with chopped eggs. Serve at once.