Combine lemon juice, avocado oil, half the salt and pepper in a shallow bowl. Whisk to combine. Separate half of the mixture into a small dish for dressing; use other portion as marinade.

Add chicken to marinade and toss to coat. Let stand 15 minutes.

Meanwhile, whisk together the arrowroot, coconut flour and remaining salt and pepper in a shallow bowl. 

Heat coconut oil in a safe nonstick skillet over medium-high heat.

Dredge the marinated chicken cutlets in the arrowroot flour mixture, then place in the preheated pan, taking care not to crowd. Cook 3 to 4 minutes per side, until golden brown and cooked through. Transfer to plate and cover. Repeat with remaining cutlets, changing the cooking oil, as needed. 

When ready to serve, toss greens with remaining dressing.

Arrange cutlets on plates, top with greens and Parmesan cheese and serve.