Make the pickles: ​​Slice cucumbers into ¼”-thick rounds and place in a large colander set over a large bowl. Lightly salt and let sit at least 1 hour but preferably overnight in the fridge. Discard liquid in bowl and pack cucumbers into a pint-sized heat proof jar. Combine 1 cup water, white wine vinegar, apple cider vinegar, sugar, 2 Tbsp. salt, mustard seeds, peppercorns, turmeric, cinnamon stick, and onions in a 4 QT Saucepan and bring to a boil over medium-high heat. Cook, stirring, until sugar and salt have dissolved, then pour hot pickling liquid over cucumbers. Let cool to room temperature, then refrigerate. Pickles will keep for about a week.

Make the balsamic onions: Heat a 4 QT Rondeau or 3.5 QT Saute Pan over medium-high heat, add a few Tbsp. of olive oil. Add onions and season with salt.  Sweat, stirring, until fully cooked (no color), 10–15 minutes. Stir sugar into onions and caramelize to golden brown, about 10 minutes more. Add balsamic vinegar and reduce until onions are a marmalade consistency, 3–5 minutes more. Transfer to a medium bowl and set aside.

Toast the buns: Prepare a grill for medium-high heat (or use your stovetop) and preheat a Carbon Steel Griddle for at least 5 minutes. Use a clean kitchen or paper towel to lightly oil Griddle, then place buns cut side down to toast. Watch carefully and remove buns once golden brown. Set aside.

Cook the burgers: Lightly form ground beef into 4 patties about 4” wide (a little wider than your buns). Oil Griddle, place patties on Griddle and season immediately with salt and pepper. Close Lid and let cook until well-browned on the underside, 3–4 minutes. Flip patties with a large heatproof spatula and season with salt and pepper. Cover and allow to cook about 2 minutes more. Place cheese slices on patties, cover, and cook until cheese starts to melt and burgers are cooked to medium rare (130–135F on an instant read thermometer), 1–2 minutes more.

Build the burgers: Spread ketchup on bottom buns and top with pickles (or serve them on the side). Place burger patties on top and finish with a spoonful of balsamic onions and the top buns.