Grate lemon rinds to equal 2 teaspoons, then halve the lemons and set aside. Combine lemon peel, oregano, oil, and garlic in a large bowl. Add chicken, season with salt and pepper and toss to cover.
Place the chicken and halved lemons on a baking tray inside the heated oven and bake for 20-25 minutes, until the chicken is cooked through and browned. Wait until slightly cooled then chop the chicken.
Heat the avocado oil in a small pan over medium heat and cook the asparagus and green onions for 3 to 4 minutes or until tender, turning occasionally. Cut asparagus and green onions into 2-inch pieces.
Divide the arugula over plates and top with asparagus and green onions. Drizzle with white wine vinegar and season with salt and pepper. Serve with chicken and lemon halves.