Preheat grill. If you are using charcoal, prepare grill for direct heat grilling. If using gas, preheat to medium-high heat.
While the grill heats up, season burgers with salt and pepper on both sides. Use your thumb to press a small divot into the center of each patty.
Make the quick burger sauce by mixing the mayonnaise, Dijon mustard, minced dill pickle, and pickle juice in a bowl. Set aside.
Clean and lightly oil your grill grates using a rag with some neutral oil. Use tongs to run the rag over the grill grates to make sure you don't get burned.
Add burgers to the grill and cook for 3-4 minutes. Flip and cook for another 3-4 minutes. One minute before the burgers are done, add cheese and Pepper and Onion Relish.
Grill buns for 1-2 minutes.
Assemble burgers with quick burger sauce, lettuce, tomato, and red onion. Serve immediately.