Prepare a charcoal or gas grill for direct grilling over medium heat. Brush and oil the grill grate.
To make the vinaigrette, add the vinegar, orange juice vinegar, Dijon, lime juice, lemon juice, basil and oregano to a blender or food processor. Pulse to blend. With the machine running, slowly add the olive oil and process until the dressing emulsifies. Transfer to a bowl. Season with salt and pepper.
Cut each romaine heart in half, leaving the core intact. Brush the cut side of the romaine halves with olive oil. Place the romaine halves, cut side down, on the grill directly over the heat and cook until the leaves develop a char and slightly wilt, about 2 minutes. Turn and cook for about another 2 minutes.
Transfer the romaine halves, cut side up, to individual plates. Spoon 2 tablespoons of the dressing over each salad and top with 1/6 of the green onions, 1/6 of the parsley and 2 tablespoons of crumbled goat cheese. Pass additional vinaigrette on the side. Serves 6.