Place eggs in a large saucepot (set carton aside). Fill with enough water to cover. Add oil. Bring just to a boil, then reduce heat to a simmer. Cook 12 minutes. Drain and run eggs under cold water until cool.Advertisement
Peel eggs. Cut a thin slice off the least pointy end to expose the yolk; if necessary, continue cutting thin slices until you see the yolk. Gently squeeze eggs to pop out yolks. Place yolks in a large bowl; return white to egg carton. Save trimmed pieces of egg white. Repeat with all eggs.
Mash yolks with mustard, dill, mayonnaise, pickle relish, salt, pepper and hot sauce. Transfer to a large resealable bag. Snip off 1 inch from a corner, pipe filling into whites, mounding over tops. There will be about 1/3 cup filling left over. Chop trimmed pieces of egg white and combine with extra filling; save for a sandwich. Close carton; refrigerate until serving.
Instead of cutting the eggs in half, slice a little off one end and fill. That way you can pop them in their carton for easy transport.