In a large skillet, heat oil or melt butter over low heat. Add onions and a generous pinch of salt. Cook over low heat for 30 to 45 minutes, stirring frequently, or until onions are very caramelized. Remove from heat and set aside.*
Butter all the slices of bread on one side. Spread the caramelized onions on the unbuttered side of four slices of bread. Top with apple slices, a slice of cheese, slices of Original Salametti, then another slice of cheese. Close the sandwiches with the four other slices of bread (buttered side up).
Grill sandwiches in a large nonstick skillet over medium-low heat until golden (to speed up caramelization, place a heavy object over the sandwiches such as a cast-iron skillet). Flip over carefully and grill until cheese is melted. Cut sandwiches in half and serve immediately.
*You can prepare caramelized onions in advance or use store-bought ones.