Fire up that grill!
In a blender or food processor add the lime juice, olive oil, garlic, cilantro or parsley, salt, cayenne, and sugar.
Process until the sauce is pale orange and the garlic is finely minced; set aside.
Remove the loosest, toughest outer leaves from the cabbage, and cut into 8 evenly sized wedges. Do not remove the stalk or inner core. Lightly brush the wedges with grapeseed or canola oil.
Place the wedges on the grill. Cover and grill until the edges of each layer are blackened and the cabbage is beginning to soften, 5 to 7 minutes per side. The cabbage is ready when it is beginning to wilt, but is still firm in the middle. (This will also be somewhat a matter of taste; I like it fairly crispy still, but you may prefer it more well-done.) If necessary, turn the heat down or move the wedges to a cooler part of the grill so they don’t burn. But don’t be afraid of those blackened edges; you want a lot of grill and char marks on the cabbage to give it smoky flavor.
Transfer the cabbage wedges to a serving platter. Brush the dressing onto the wedges and serve, with wedges of lime to garnish.