1 tablespoon extra-virgin olive oil
3 cups very thinly sliced red onion (from 2 medium red onions)
2 tablespoons granulated sugar
3/4 teaspoon salt, divided
2/3 cup Cabernet Sauvignon wine
1 tablespoon balsamic vinegar
1 1/4 pounds lean ground beef
1/2 teaspoon salt
1/4 teaspoon black pepper
2 ounces thinly sliced gruyere cheese
4 (2-ounce) sourdough or Kaiser rolls
Heat the oil in a large skillet over medium heat. Add the onion, sugar, and 1/4 teaspoon of the salt and cook, stirring occasionally, until the onion is very soft, about 15 minutes. Add the wine and vinegar and cook until the liquid is nearly gone, 12 to 14 minutes longer (25 to 30 minutes total). Cover and set aside.
Prepare a grill for direct-heat cooking (400° to 500°F). Brush the grates lightly with oil.
Mix the beef, remaining 1/2 teaspoon salt, and pepper with a fork; form into four 3/4-inch-thick patties. Press your thumb in the center of each burger to form an indentation. Grill the patties 4 to 5 minutes, turn and grill another 4 minutes, and top with the cheese. Cook until the cheese is melted and the burger is cooked to desired degree of doneness (160° for medium).
Place the burger on the bottom half of the roll; top with the onion jam and close the sandwich.