Mix ground beef, chopped sun-dried tomatoes, minced garlic and onion, dried basil, ground cumin, 1/4 teaspoon sea salt, and 1/8 teaspoon black pepper in large glass bowl. Cover with plastic wrap and refrigerate at least 1 hour. Can be prepared 4 hours ahead. Keep chilled.
Prepare BBQ Grill: set to medium-high heat.
Heat reserved tomato oil in heavy large skillet over medium-high heat. Add thinly sliced onions and saute until soft and starting to brown, about 5 minutes. Add balsamic vinegar. Sprinkle with sea salt and black pepper. Simmer until onions are deep brown, stirring occasionally, about 10 minutes. Remove from heat.
Please note: Can be prepared 1 hour ahead. Let stand at room temperature. Wrap in foil and reheat on grill 10 minutes prior to service.
Grill burgers until cooked through, about 5 minutes per side. Top with slices of Monterey Jack if desired. Arrange burger on your favorite bun or atop seasonal greens. Top with balsamic onions and enjoy.