Pulse 1/2 cup oats in a food processor until finely ground. Add cheese, parsley, 1/4 teaspoon each salt and pepper, and remaining 1/4 cup oats. Pulse just to combine. Transfer mixture to a shallow bowl.
Remove tenders from chicken; reserve for another use. Pound chicken to a 1/2-inch thickness. Coat with 1 tablespoon olive oil; season with salt and pepper. Cover chicken in oat mixture, pressing lightly to adhere.
Heat 2 tablespoons oil in a medium nonstick skillet over medium-high. When it shimmers, add half of chicken in a single layer and cook, flipping once, until golden brown and crisp, 5 to 6 minutes. Transfer to a wire rack. Wipe skillet clean and repeat process with 2 more tablespoons oil and remaining chicken. Serve with greens.