To make salad: In a large saucepan, combine beets, vinegar, l teaspoon salt, and enough cold water to cover beets by 2 inches; bring to a boil.
Reduce heat and cook until tender, 40 to 45 minutes. Cool beets in cooking liquid; peel. Using a handheld slicer, cut beets into 1/8-inch-thick slices. Set aside.
Meanwhile, heat a grill or a grill pan. Brush both sides of onion slices with olive oil; sprinkle with pepper and remaining 1/2 teaspoon salt. Grill onions over medium heat until evenly charred and tender, 2 to 3 minutes per side.
To make vinaigrette: In a small bowl, whisk together vinegar, brine, honey, oregano, salt, and pepper. Gradually add oil, whisking in a thin, steady stream until blended.
Arrange 1 cup greens and 1/4 cup cheese on each of 6 plates. Divide the sliced beets and grilled onions equally among the plates and drizzle with vinaigrette. Finish with sprinkle of dill.