If chicken breasts are uneven, pound to an even ¾-inch thickness between plastic wrap. Pat completely dry with paper towels.
Sprinkle kosher salt evenly over both sides of chicken. Place on a wire rack over a baking sheet and refrigerate uncovered for 30 minutes to 2 hours.
Combine smoked paprika, garlic powder, onion powder, black pepper, dried oregano, and cayenne in a small bowl.
Place cast iron grill pan in oven and preheat to 425°F for 20 minutes.
Remove chicken from refrigerator. Pat dry again, brush with avocado oil on both sides, then apply spice rub evenly.
Carefully remove the preheated grill pan from oven and place over medium-high heat for 2-3 minutes.
Add a teaspoon of oil to the pan. Place chicken breasts at a 45-degree angle to the ridges and sear undisturbed for 3-4 minutes.
Rotate chicken 90 degrees (same side) and continue cooking for 2 minutes to create crosshatch grill marks.
Flip chicken and repeat the searing process on the second side.
Transfer the grill pan with chicken to the oven. Continue cooking for 4-6 minutes, until internal temperature reaches 160°F.
Remove chicken, transfer to a cutting board. Tent loosely with foil and rest for 5 minutes before slicing.