Heat the oven to 200°C (400°F) and line a large sheet pan with parchment.
Trim the ends of the squash, then cut it crosswise into two halves. Scoop out the seeds and stringy pulp with a spoon. Brush the cut sides with 1 tablespoon of the olive oil and season with salt and pepper.
Place the halves cut-side down on the sheet. Roast for 35 to 40 minutes, until the skin gives when pressed and a fork slides through the flesh with light resistance. The cut edges should be lightly golden.
In the last 12 minutes, toss the cherry tomatoes with the remaining oil, salt, and pepper on a second small pan and roast until their skins blister and split.
Whisk the dressing ingredients together in a small jar until emulsified and slightly thick. Taste and adjust the lemon and salt.
Rest the squash for 10 minutes, until you can hold it comfortably. Scrape the flesh with a fork lengthwise into strands, then spread it across the sheet pan for 5 minutes so the steam can escape.
Divide the cooled squash among four containers. Add the chicken and roasted tomatoes in separate sections. Keep the spinach, feta, seeds, and dressing in separate small containers or a top layer.
To eat, reheat the squash and chicken together until steaming, then add the spinach, feta, and seeds, and pour over the dressing.