Mix all ingredients for the patties together and divide into ⅓ cup portions. Press each portion into a thin patty. Cook on a griddle or large frying pan until cooked through or done to your liking- about 3 minutes per side. Add a slice of cheese on top towards the end of cooking to let melt.
Butter 1 side of each piece of bread and cook along side the patties until golden brown and toasted.
Combine mayo, ketchup, relish, and dill weed.
Assemble the burgers with one patty on a piece of toast topped with tomato and lettuce. Spread about a tablespoon of the sauce on another slice of the toast and top the sandwich with it. Serve hot!