Heat grill to about 450 degrees F.
Meanwhile, make pesto: Add baby arugula to the bowl of a food processor and pulse about 20 seconds, until leaves are broken down. Stop to scrape down sides, as needed.
Add walnuts, peeled garlic cloves and zest and juice of 1 lemon into the bowl. Pulse for another 10 to 15 seconds.
Add in ½ teaspoon kosher salt and ¼ teaspoon cracked black pepper. With the lid secured, turn the food processor on low and slowly drizzle in ½ cup olive oil.
Taste and adjust salt and pepper, as needed. Set aside.
Peel off outer layers of cabbage and slice into ½-inch rounds. Place on a large baking sheet and brush cut sides with olive oil, and season with salt and pepper.
Carefully place cabbage rounds onto preheated grill and cook for 5 minutes per side. Add grilled cabbage rounds to a serving platter and smear a large spoonful of Arugula Pesto onto each round.
Serve alongside the protein of your choice. Store leftover pesto in a jar with air-tight lid for up to a week in the refrigerator.