For ratatouille, warm oil in a large nonstick skillet over medium heat. Add the onion, garlic, eggplant, zucchini, salt, and pepper. Cook 8 minutes, stirring, until softened and cooked down. Stir in tomato, red bell peppers, and vinegar; cook 10 minutes, until thickened. Remove from heat; stir in basil. Set aside.
For burgers, combine ricotta, Worcestershire, mustard, basil, salt, and pepper in a large bowl. Add turkey; mix with clean hands to combine. Divide into 4 pieces; lightly form each into a 1-inch-thick patty. Grill patties over medium-hot coals 7 minutes per side, turning once.
Place burgers and ratatouille between buns, and serve. Leftover ratatouille will keep, tightly covered, for a week in the fridge.