OK people, settle a mrs sage and my argument

Sandwiches were named after John Mantagu (1718-1792), the Fourth Earl of Sandwich.  The story goes that he was a gambler, who wanted meat between two slices of bread, so he would not have to leave the table.  John Mantagu was a gambler but also enjoyed playing cards in a non-gambling setting.  He also was a very industrious worker and often ate meals at his desk.  It seems that he was so fond of eating bread and meat in between two slices of bread that everyone in his circle started to refer to the manner of eating bread and meat in this manner as a sandwich.  This was true as of 1765, when the Earl of Sandwich was 47/48.  By that time, he has already helped negotiate the 1748 treaty of Aachen (England traded Madras for Louisbourg, which it would recapture and hold beginning in 1758, and received an unspecified payment from Spain, which ultimately amounted to £10,000, pursuant to the Treaty of Madrid), served two spells as First Lord of the Admiralty (basically, the Secretary of the Navy), and Secretary of State for the Northern Department.  In this context, North refers to Northern (Protestant) Europe, as opposed to Southern (Catholic and Muslim) Europe.  The Northern Secretary of State was subservient to the Southern Secretary of State, whose work was viewed as more important.

As the English originally used the word, a sandwich is made from bread sliced from a loaf.  This is the original meaning in the Australia, as well.  Outside of America, a burger refers to the type of meat.  If you order a burger, you may get just the meat.  A burger on a split roll is a burger but not a sandwich outside of America.

A hot dog is similar.  Outside of America, a hot dog is a method of serving meat.  In England, if you order a hot dog, you get a version of sausage in some kind of roll, but it is usually not what you envision a hot dog is in America.

In American parlance, a sandwich typically does not need to be made from a larger loaf of bread.  It can be made from a "split roll."  Thus, in America, according to the definition, burgers and hot dogs, served in a split roll, are "sandwiches." In Australian and English parlance, there is a difference and neither a burger nor a hot dog is a "sandwich."

My answer to you is "yes," both hamburgers and hot dogs in a split roll are "sandwiches" according to the American definition of the word.  Outside of America, it gets murkier.