Heat 1 tablespoon of the olive oil and the butter in a large, deep skillet over medium heat. Once the butter has melted and the olive is shimmering, place the onions in the skillet cut-side-down. Don't disturb the onions for a minute or two. Once the onions are golden on one side, give them a toss and sprinkle them generously with salt. Push the onions to the side of the skillet and add the garlic. Sauté until fragrant, about 2 minutes.
Bundle the thyme sprigs together to form a bouquet. Secure the thyme with butcher's twine and add it to the skillet. Pour in the wine and bring the mixture to a boil. Reduce the heat to a simmer and cover. Let the onions braise for 20 minutes or until very tender. Remove the onions from the heat and keep warm until ready to serve.
While the onions are braising, form the ground beef into four evenly-sized patties. Transfer the patties to the fridge to chill until ready to cook. Give them at least 15 minutes in the fridge before cooking.
Heat the remaining olive oil in a large cast-iron skillet over medium-high heat. Sprinkle the chilled burger patties with salt and pepper on one side. Place the patties in the skillet, seasoned side down. Create a dimple in the center of each patty and season the naked side with salt and pepper. Cook the burger for 2-3 minutes, then flip. Add a slice of Oka to the top of each burger and cook for 2-3 minutes more. The burgers should be tender and soft pink in the center. Cook for longer if you prefer a burger that is more well done.
While the burgers are cooking, place all of the Patty Sauce ingredients in a small bowl and whisk to combine.
Slather patty sauce on two pieces of toasted rye bread. Add 3-4 pickle slices to one slice and place a burger patty on top. Pile a good amount of the braised onions on top of the burger and place the remaining slice of rye bread on top. Repeat until you have four patty melts.
Serve the sandwiches immediately with a side of fries and soda for the true diner experience.
