Create Fillets: Slice the chicken through the equator to create 4 fillets. Cover them with plastic wrap and pound with a meat mallet, the side of a can, or a rolling pin to an even thickness of about ½ inch.
Season Chicken: Whisk the Blackening Seasonings together in a small bowl. Pat the chicken fillets dry, then lightly brush both sides with olive oil. Generously season all over with the spice mixture, pressing it into the chicken so it adheres.
TO GRILL
Preheat the grill to 400°F, and clean and grease the grates. Grill the chicken undisturbed for 3-4 minutes per side, or until the chicken reaches 160°F on an instant-read thermometer. In the last minute of cooking, place the cheese on the chicken and cover to melt it.
Transfer the chicken to a cutting board and let it rest for at least 5 minutes to reach 165°F.
TO PAN SEAR
Melt 2 tablespoons butter with 2 tablespoons oil in a large, heavy-bottom skillet over medium-high heat. Once hot, add the chicken and cook for 3-4 minutes per side, until blackened, and the chicken reaches 160°F on an instant-read thermometer. In the last minute of cooking, place the cheese on the chicken and briefly cover the skillet to melt.
Transfer the chicken to a cutting board and let it rest for at least 5 minutes to reach 165°F.
SANDWICHES
Toast the buns: Generously brush the cut sides with softened butter. Toast buns in a skillet or on the grill (cut side down) until golden.
Assemble: Top the buns in this order: lettuce, tomato, chicken, Caesar dressing, red onion, avocado. Dig in!
Notes
Chicken: Look for thick chicken breasts so you can easily slice them in half horizontally, but not so large/long that they dwarf the buns.