Put bulgur in a large bowl. Cover with 1 cup boiling water and let soak until water is absorbed, 20 to 30 minutes.
Heat grill to high (450° to 550°). Brush chicken with 1 tbsp. oil and sprinkle on both sides with half the salt, pepper, and paprika. Grill, turning once, until cooked through, about 5 minutes; set aside.
Add remaining oil, salt, pepper, and paprika, plus all other ingredients except feta, to bowl of bulgur and toss to combine.
Slice chicken. Divide salad among 4 plates and top with chicken and feta.
*Bulgur--steamed crushed wheat--comes in three sizes. We like the texture of medium, but you can use another size; just adjust the soak time accordingly.