Place two oven racks in the top third and center of the oven and heat the oven to 400 degrees.
In a small bowl, toss the shallots with the apple cider vinegar. Set aside to marinate.
Pat the chickpeas completely dry with paper towels, then spread them out on a large rimmed sheet pan. Drizzle with 2 tablespoons of the olive oil, sprinkle with salt and toss to coat. Spread out in a single layer.
On a second large rimmed sheet pan, add all three types of squash. Drizzle with the remaining 4 tablespoons olive oil and sprinkle with a good pinch of salt and a light dusting of cayenne. Toss to coat, then arrange in a single layer. (Some squash can go on the sheet pan with the chickpeas, if needed for space.)
Place the chickpeas on the top rack and the squash on the center rack of the oven and roast for about 30 minutes, until the chickpeas are beginning to crisp and the squash is very tender. Remove the chickpeas from the oven. (If you roasted some of the squash with the chickpeas, transfer it to the pan with the rest of the squash.) Sprinkle the paprika over the top of the chickpeas and shake the sheet pan to coat. Set aside to cool and finish crisping on the sheet pan. The chickpeas can be cooled and stored at room temperature in an airtight container for up to 24 hours.
Turn on the broiler and move the squash to the top oven rack. Cook for 3 to 5 minutes, until nicely charred. Use tongs to pile the squash onto a serving platter.
Scatter the chickpeas over and around the squash. Drain the shallots and sprinkle them over the squash, along with the feta and mint, finish with a couple more pinches of cayenne before serving. This recipe can be served warm or at room temperature.