In a medium bowl, mix mayonnaise, ketchup, gherkins, gherkin juice, mustard, Worcestershire, salt, paprika, pepper, and garlic powder until combined. Cover and refrigerate until ready to use.

In a large cast-iron or nonstick skillet over medium-high heat, melt 1 tablespoon butter. Add yellow onions, sugar, and a pinch of salt and stir to combine. Cook, stirring occasionally and adding a splash of water if onions brown too quickly, until softened and golden brown, about 10 minutes. Transfer onions to a large bowl and let cool slightly. Reserve skillet (do not wipe out).

Add chicken, panko, mayonnaise, Worcestershire, and garlic powder to bowl with onions; season with salt and pepper. Using your hands or a wooden spoon, gently mix to combine, being careful not to overmix.

Shape chicken mixture into 6 patties about ½" thick. Using your thumb, make an indentation in the center of each patty (this prevents the burgers from shrinking too much while they cook).

In reserved skillet over medium-high heat, heat oil and 1 tablespoon butter until very hot and butter is bubbling, about 2 minutes. Cook 3 patties, turning halfway through, until seared and an instant-read thermometer inserted into the center registers 165°, 8 to 10 minutes.

Place 1 slice of cheese on top of each patty, cover, and continue to cook until cheese is melted, 30 to 60 seconds. Transfer to a plate and tent with foil. Repeat with remaining 1 tablespoon butter (no need to add more oil) and 3 patties.

Spread burger sauce on cut sides of buns. Top bottom buns with lettuce, burgers, tomato, and red onion. Close with top buns.