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Arrange racks in middle and upper thirds of oven; preheat to 450°F. Cut squash in half crosswise where the neck and body meet. Peel and seed. Slice neck lengthwise into ¼"-thick planks (about 2 ½" long). Slice body into ¼" pieces.
Divide squash, oil, thyme, garlic, 1 tsp. salt, and ½ tsp. pepper between 2 rimmed baking sheets. Toss to combine, then arrange in a single layer. Bake, tossing once halfway through and rotating trays, until squash is tender, 18–20 minutes.
Meanwhile, bring vinegar, sugar, remaining ¾ tsp. salt, and ½ cup water to a boil in a small saucepan. Remove from heat, then stir in onion. Let cool.
Whisk mayonnaise and mustard in a small bowl. Remove thyme leaves from stems, then finely chop leaves and garlic. Stir into mayonnaise mixture.
Slice rolls in half and toast in oven 5 minutes. Spread 2 Tbsp. mayonnaise mixture on bottom halves. Top each with squash, cheese, 1 ½ Tbsp. pickled red onion, and arugula. Serve with chips alongside, if using.
Squash can be roasted 3 days ahead; cover and chill. Onions can be made 1 month ahead; cover and chill.