Cut the squash in half from stem to end. Scrape out the seeds with a spoon. Cut into approximately 1-inch chunks. Place on a rimmed baking sheet, drizzle with 1 tablespoon olive oil, sprinkle with salt and pepper and roast for 30 minutes, or until fork tender and starting to brown, stirring once or twice while cooking.
Combine the remaining olive oil with the vinegar, 2 tablespoons maple syrup, the mustard, ½ teaspoon salt, and ¼ teaspoon pepper. Whisk well.
When the squash is done, drizzle with 2 tablespoons of the dressing and 1 tablespoon maple syrup, toss to coat, and let cool slightly.
Toast the pecans in a dry skillet over medium heat until warm and fragrant, stirring often, about 5 minutes.
Cut the beets into chunks and toss with 1 tablespoon dressing.
Toss the arugula with just enough dressing to lightly moisten. Arrange the arugula on a platter, top with the squash, beets, pecans, and cheese. Serve the remaining dressing on the side.