Preheat broiler. Remove the core from both cabbages. Roughly shred cabbages with a knife and place in a large bowl.
Toss cabbage with olive oil and spread half of it on a roasting sheet. Broil, for about 5 minutes, until well wilted and slightly charred. Repeat with remaining cabbage.
Return all roasted cabbage back to the bowl and toss with lemon juice, Parmesan, and parsley. Taste for seasoning and add a sprinkling of salt, if you like.
Transfer to a serving bowl or platter and garnish with croutons. Serve warm or at room temperature.