Place sliced onions in a pot. Cover onions about 90% with cola and add salt.
Place over medium heat and reduce until cola is completely reduced and evaporated. Be careful not to burn. Set aside.
Combine cheese and water in a saucepan. Heat slowly and whisk until well incorporated. Set aside.
Form three burger patties, about 4oz each.
On a preheated GrillGrate Rails Down, griddle patties over medium-high heat, evenly smash the patty flat when it hits the grill, and start to toast the rye bread.
Flip the patties over and top with cheese, place patty on toasted rye bread, and top with onions.