Combine egg, bread crumbs, minced vegetables, and seasonings in a mixing bowl. Add ground beef and mix thoroughly. Form into 3/4-inch balls. Heat oil and butter in a medium fry pan. Add meatballs; shake pan and turn meatballs to brown on all sides and cook through, about 5 minutes. Cool meatballs. This may be done ahead of time and meatballs refrigerated until needed.
Roll meatballs in barbeque sauce. Place in center of won-ton wrapper. Bring corner over meatball. Brush remaining corners with lightly beaten egg white. Wrap meatballs by bringing corners up and sealing them to each other so it resembles a wonton. Continue until all meatballs are wrapped. Deep fry 3 or 4 wrapped meatballs at a time for 2 to 3 minutes, turning to brown on all sides. Drain on paper towels.