Remove chicken and from refrigerator and let come to room temperature. This will reduce cooking time to about 6 minutes from 8, if cold.
Remove arugula from refrigerator to take edge off and make more flavorful. Slice sweet pepper into thin strips.
Place chicken on parchment or in plastic bag and pound with meat hammer– I like a heavy metal one ($12.99 Sur La Table)– until about 1/2 inch thick. Any thinner and the meat tends to dry out. Use parchment paper, plastic produce bag or gallon zip lock layered over the pounding surface for sanitization.
Turn heat onto medium high and add 3 tablespoons of coconut oil into a large cast iron pan.
Dry chicken pieces with paper towels and place slowly in hot oil. Add salt pepper and half of the za ‘atar. Cook for three minutes and turn chicken over with tongs. Repeat salt pepper and za’atar, and cook until done–about 3 more minutes. Check by slicing thickest part of a breast to make sure not pink. Remove from heat and let chicken rest to reabsorb juice.
To plate, layer chicken, a mound of arugula, and top with the sweet peppers. Squeeze fresh lemon juice over arugula and serve extra lemon slices.
If desired, the arugula can be lightly dressed with a lemon vinaigrette. Arugula leaves bruise very easily so pour vinaigrette around rim of the bowl and lightly move arugula around to incorporate (thank you, Bobby Flay for this technique).