Marinate the chicken: Lightly pound the chicken into an even, 1/2-inch thickness (I place a sheet of plastic over the top first for easy cleanup). Place in a large ziptop bag or a shallow dish for marinating.
Add the oil, lemon juice, honey, Dijon, smoked paprika, garlic powder, salt, and pepper. Toss to evenly coat the chicken. Let marinate at room temperature for 30 minutes, or place in the refrigerator for up to 6 hours. (If marinating for more than 30 minutes, rotate the chicken once or twice throughout.)
When ready to cook, heat an outdoor gas or charcoal grill (or an indoor grill pan) to medium-high, about 375°F. Grill the chicken for 8 to 12 minutes, flipping halfway through. When the chicken is nearing the end of its cooking time, place a slice of cheese on top to melt it. The chicken is done when it reaches 155 to 160°F on an instant read thermometer (chicken is considered safe to eat at 165°F, but its temperature will rise as it rests. DO NOT overcook or the chicken will be dry).
Let rest for 5 minutes.
If desired, place the buns cut-sides down on the grill to toast (or toast them briefly under the oven’s broiler, cut-sides up).
Assemble the sandwiches: spread the cut side of the bottom bun with Greek yogurt and/or Dijon mustard and/or mayo. Top with lettuce and tomato. Sprinkle lightly with salt and pepper, then add the chicken and onion (or place the chicken on first; just know the bottom bun will get a little soggy). Add the top bun and onions if desired. Enjoy immediately.
TO STORE: Refrigerate the grilled chicken in an airtight storage container for up to 4 days. Store the buns at room temperature.
TO REHEAT: Gently rewarm chicken on a baking sheet in the oven at 350°F or in the microwave. Toast buns in the oven as well if desired.
TO FREEZE: Freeze the grilled chicken in an airtight, freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating. Freeze buns in an airtight freezer-safe ziptop bag. Thaw at room temperature or toast from frozen.