Place a box grater over a few layers of paper towels. Grate the white onion onto the paper towels, then gather the towels and gently squeeze out excess moisture. Set aside.

Using the same grater, grate the cold butter into a small bowl.

In a large mixing bowl, combine the grated onion, grated butter, ground chicken, mayonnaise, Parmesan cheese, garlic, Worcestershire sauce, Dijon mustard, lemon zest, salt, and pepper. Mix gently until just combined—do not overwork.

Line a baking sheet with parchment paper. Divide the mixture into 4 equal portions and shape into patties about 1/2 to 3/4 inch thick. Press a shallow indent in the center of each patty with your thumb. Place the baking sheet in the freezer for 20 minutes to firm up the patties.

Clean the grill grates and preheat your grill to medium-high heat (about 400°F).

Place the chilled patties directly over the heat. Close the lid and cook for 5 to 6 minutes per side, or until the internal temperature reaches 165°F on an instant-read thermometer. If using cheese, lay a slice of provolone on each patty during the last 1 to 2 minutes of cooking. Remove from the grill and let rest for 5 minutes.

While the burgers rest, butter the cut sides of the buns. Place them butter-side down on the grill and cook for 30 to 60 seconds, until lightly toasted with grill marks. Remove carefully.

To build each burger: Place a romaine leaf on the bottom bun. Drizzle with a little Caesar dressing, then set the chicken patty on top. Add more dressing, a few slices of red onion, another romaine leaf, a sprinkle of shaved Parmesan, and a handful of crushed croutons. Cap with the top bun.

Serve immediately with your favorite sides, like fries or a simple salad.