For the sweet Greek dressing: Whisk together the vinegar, honey, mustard, oregano and garlic in a bowl. Slowly stream in the olive oil while whisking. Add salt and pepper to taste. Measure out 1/2 cup of the dressing for marinating the chicken. Set aside the remaining dressing for drizzling on the salad.
For the chicken: Put the chicken in a nonreactive container, pour the reserved 1/2 cup sweet Greek dressing over it and toss to coat. Marinate the chicken, refrigerated, for 2 hours.
Preheat a grill to medium heat.
Oil the grill grates with vegetable oil. Grill the chicken until golden on each side and the internal temperature registers 165 degrees F, 7 to 8 minutes per side. Let rest before slicing.
For the salad: Brush each romaine half with vegetable oil and season with salt and pepper. Grill on high on the cut sides until slightly charred, 2 to 3 minutes.
Place a seared romaine half on a serving plate. Line up some tomatoes, feta, olives, cucumbers, onions and sliced chicken atop the romaine. Repeat with the remaining ingredients. Drizzle with the sweet Greek dressing. Garnish with cracked black pepper and sea salt and serve.