Preheat an oven to 425°F.
Trim off the stem end of each delicata squash. Carefully cut each squash in half lengthwise, then remove the seeds. Slice each half into 3/4-inch thick half moons. Place chicken thighs on a rimmed baking sheet and drizzle with 1 tablespoon olive oil. Using your hands, make sure the thighs are coated on both sides with olive oil. Season both sides with well with salt and pepper. Scatter thyme sprigs over half the baking sheet. Place seasoned chicken thighs on top of the thyme sprigs, spaced out so they aren't touching.
Place delicata squash on the other half of the sheet pan. Drizzle with remaining olive oil, and season with salt and pepper. Roast chicken and squash for 25 to 30 minutes until the squash is tender and the chicken registers an internal temperature of 165°F. Pop the sheet pan under a broiler set on high for about 5 minutes to get some golden color on top. Let chicken rest for 5 minutes before serving.
While the chicken is cooking, prepare the vinaigrette. Place vinegar, garlic, mustard and maple syrup in a medium bowl with a pinch of salt and crushed red chile flakes, whisking to combine. Let sit for a few minutes, then whisk in olive oil. Season to taste with additional salt.
To serve, divide chicken thighs between 2 to 4 plates. Then, make a salad on the side with the arugula, roasted squash, pumpkin seeds and a sprinkling of parmesan. Drizzle both salad and chicken with the vinaigrette. Enjoy immediately.