Preheat the oven to 425°F.
Melt the butter over medium-high heat in a medium-sized skillet (preferably not non-stick). Add onions and salt and stir to coat, then turn heat down to medium-low. Refrain from stirring for the first 10 minutes; you want your onions to stick to the bottom of the pan a bit to coerce the sugar out. Once you start to see brown edges (fond), stir occasionally, scraping the bottom of the pan, and cook until caramelized, 20-30 more minutes. Add the thyme and white wine and cook for 3-5 minutes until liquid has evaporated.
Meanwhile, slice the squash lengthwise into thin wide strips - aim for ⅛-inch thick, but it's okay if you can't manage this consistently (if you’re confident with a mandoline slicer, now's the time to use it). Place onto a baking sheet and add oil, salt, pepper and cheese and toss the mixture with your hands so everything is evenly dispersed. Let sit for 5-10 minutes, until the squash is nice and pliable.
Put the caramelized onions in the bottom of a round 8-inch baking dish (or divide them evenly among a couple of smaller individual baking dishes). Using the thinnest, flimsiest pieces of squash first, tightly roll a piece of squash and put it in the center of the dish. Working outward, continue to curl and coil each individual piece until you reach the sides. Reserve the thickest, least pliant pieces for the outer edges. Top with any cheese that may have stuck to the baking sheet.
Bake uncovered for 25-30 minutes, until cheese is brown and bubbling and squash is tender. Serve immediately, using tongs to place full ribbons of squash onto your plate.