Combine the olive oil, red wine vinegar, garlic, lemon slices, rosemary, brown sugar, and 1/4 teaspoon kosher salt in a small bowl. Whisk until the brown sugar fully dissolves. Then arrange the cauliflower wedges in a 9x13-inch baking dish and pour the marinade over top. Cover and let sit at room temperature for 30 minutes or in the refrigerator for 30-90 minutes. Rotate the wedges 1-2 times while marinating to ensure even coverage.
When approaching grill time, preheat your outdoor grill to high heat (500-550°) and generously coat the grates with vegetable oil or nonstick spray.
Place the marinated cauliflower wedges directly on the hot grill (cut side down) and reserve the marinade in the baking dish. Close the lid and let the wedges cook undisturbed on one side for 6-7 minutes or until they develop dark grill marks. Then flip and do the same on the other cut side (another 6-7 minutes). When done, return the wedges to the baking dish with the marinade.
Arrange the grilled cauliflower wedges on a large serving plate, sprinkle with the remaining 1/2 teaspoon kosher salt, and spoon the reserved marinade and lemon slices over top. Finish with a generous sprinkling of grated Parmesan or Pecorino Romano cheese. Serve warm or at room temperature.