For the caramelized onions: Add the oil to a large skillet set over medium-high heat. Add the onions and 1 teaspoon salt and cook until the onions have softened and are starting to brown in spots, about 10 minutes, stirring occasionally. Reduce the heat to medium and continue cooking until the onions are deeply browned and caramelized, 40 to 50 more minutes, stirring frequently toward the end of the process. Season with salt if needed. Transfer to a small bowl.
For the turkey burgers: Line a baking sheet with parchment paper. Combine the mayonnaise, Worcestershire sauce, garlic powder, onion powder, 2 teaspoons salt and 1 teaspoon pepper in a large bowl. Stir to combine, and then add the turkey and mix until just combined; do not overmix or the meat will get tough. Lightly wet your hands then divide the mixture into 4 equal portions. Pat into 4-inch wide patties (about 3/4-inch thick) and put on the prepared baking sheet.
Heat the oil in a large nonstick skillet over medium heat until hot. Add the turkey patties (can be done in batches if needed) and cook until browned on the first side, 4 to 5 minutes. Flip and top each patty with a slice of cheese. Cover with a lid or baking sheet until the cheese is melted, the burgers are browned on the second side and the internal temperature is 165 degrees F on an instant-read thermometer, 4 to 5 more minutes.
To serve: Spread 2 tablespoons of the Cranberry Relish on the bottom half of each bun. Layer each with 3 bread and butter pickles, a patty and 1/4 of the caramelized onions. Top with the other half of bun. Serve immediately. Any leftover cranberry relish can be kept in the refrigerator in an airtight container for up to 2 weeks.
Add the cranberries, apple, orange zest and juice, sugar and 1/2 teaspoon salt to a food processor and pulse until the texture is like a chunky pesto. Transfer to a small bowl and refrigerate while you prepare the other burger components.