Preheat the oven to 400 degrees F. Line a sheet pan with parchment paper. Scatter the handful of thyme sprigs across the paper. Rub the halves of squash with 1 tablespoon of olive oil, sprinkle with salt and pepper and lay face down on the thyme sprigs. Place squash in the oven and roast until very tender, about 30 minutes. Remove from the oven and cool.
Once you can handle the squash, scoop the cooked flesh from the skins into a bowl, discarding the thyme sprigs. Set aside.
In a large soup pot, heat the remaining 2 tbsp olive oil over medium heat. Add the garlic, thyme and chili flakes, stirring constantly until fragrant, about 1 minute. Add the roasted squash and stir it all up, mashing the squash down into the sautéing aromatics. Add the lemon juice and the vegetable stock, stir until roughly combined and remove from the heat. Puree the squash and garlic mixture in a blender or food processor, adding more liquid if necessary.
Return the squash sauce to the soup pot over medium heat. Season with a fat pinch of salt and lots of pepper at this point.
Cook pasta according to package directions. Reserve 1/4 cup of cooking water before draining.
Once squash sauce is simmering, add cooked pasta, chickpeas, pecorino, arugula and pine nuts, reserving a bit of each for garnish if you like. Add some of the pasta water of you want to loosen the mixture up a bit. Serve hot.