Place a box grater over a few layers of paper towel. Grate the white onion onto the paper towel, then gather the towel and gently squeeze out excess moisture. Using the same grater, grate the cold butter directly into a small bowl and set aside.

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

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

Clean the grill grates and preheat your grill to medium-high heat (about 400°F to 450°F).

Place the frozen patties on the hot grill over direct heat. Close the lid and cook for 5 to 6 minutes per side, or until an instant-read thermometer inserted into the center reads 165°F. During the last 1 to 2 minutes, top each patty with a slice of provolone cheese. Remove the burgers to a plate and let rest for 5 minutes.

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

On each bottom bun, layer a piece of romaine lettuce, a drizzle of Caesar dressing, the grilled chicken patty, another drizzle of Caesar dressing, a few slices of red onion, another lettuce leaf, shaved parmesan, and crushed croutons. Cap with the top bun.

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