How to Make It
Preheat broiler; place a rack close to heat source. Line 2 large baking sheets with foil and brush lightly with oil. In a large ziplock bag, mix together 1 Tbsp. balsamic vinegar, 1 tsp. rosemary, garlic and 1/2 tsp. each salt and pepper. Add steak; seal bag and rub marinade into steak. Let stand at room temperature for 15 minutes.
Arrange eggplants, peppers and onion in a single layer on baking sheets; brush tops with 2 1/2 Tbsp. oil. Sprinkle with remaining 2 tsp. rosemary and 1/2 tsp. each salt and pepper. Broil in batches, turning once, until tender and golden brown on both sides, about 10 minutes per sheet.
In a medium nonstick skillet, warm remaining 1/2 Tbsp. oil over medium heat. Remove steak from ziplock bag; discard marinade and cook steak, turning, 6 to 8 minutes total (depending on thickness) for medium-rare. Transfer to a cutting board and let rest for 5 minutes before slicing.
Stir together cheeses and divide among 4 plates, spreading in center. Top each with roasted vegetables and sliced steak. Drizzle with remaining 1 Tbsp. balsamic vinegar. Sprinkle with basil, season with pepper and serve.