Heat olive oil at medium-high in a large skillet and add the sliced onions. Sauté, stirring occasionally, until onions are translucent and begin to brown on the sides, about 10 minutes.
Reduce heat to medium-low and continue cooking, stirring occasionally, until onions become deep brown and are caramelized, about 40 minutes. Add sea salt to taste. Note: if onions begin to stick to the skillet at any point, add a few tablespoons of water to de-glaze the pan.
Add ground beef, oregano, parsley, paprika, and sea salt to a mixing bowl. Use your hands to mush everything together until ingredients are well-combined. Form three 1/3-pound patties, or four ¼-pound patties.
Preheat grill to high heat. Carefully place patties on grill and cover with the grill lid. Grill for 3 minutes until first side is golden-brown and slightly charred. Flip patties, cover, and cook 2 minutes.
Add the sliced cheese to the burgers and cover. Cook until cheese has melted, about 1 minute. Note: for well-done burgers, cook an additional 2 minutes. (Note: See instructions below for cooking burgers in a skillet).
Spread just 2 to 3 teaspoons of Harry & David Country Cranberry Relish on each slice of toasted burger bun. Top burgers with caramelized onions, and serve alongside your favorite side dishes.