Heat grill or grill pan to medium-high. In large bowl, combine lemon juice, 1 tablespoon oil, oregano, garlic powder, Aleppo pepper, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Add chicken and toss to coat. Let sit 3 minutes (resist marinating longer, as the lemon juice could start cooking the chicken).

In second large bowl, toss kale with remaining 2 tablespoons oil and 1/4 teaspoon each salt and pepper.

Grill chicken until charred and cooked through, about 2 minutes per side; transfer to plates. Grill kale, turning often, until charred and just tender, 3 to 4 minutes total; transfer back to bowl. Grill lemons, cut sides down, until charred, 1 to 2 minutes.

Squeeze half of grilled lemon over kale and toss to combine, then toss with Parmesan. Serve over chicken with additional shaved Parmesan and grilled lemon halves.