Adjust oven rack to center position and heat oven to 425 degrees. Cut each squash in half through the root end, then use a spoon to scoop out the seeds. Cut each half into 2-inch-thick wedges.
Spread out the squash on a large rimmed baking sheet and drizzle with ¼ cup of oil. Season with salt and pepper and toss until well coated. Roast, flipping once, until golden brown and very tender, 25 to 30 minutes.
Meanwhile, combine the shallots, garlic and remaining ½ cup olive oil in a small saucepan or skillet over medium heat. Cook, stirring often, until the shallots begin to shrivel, crisp and turn golden brown, 4 to 6 minutes. Immediately remove from heat and sprinkle in the sesame seeds, paprika, crushed red pepper and ½ teaspoon salt. Let cool for 10 minutes.
Using a microplane, zest the entire lemon and half the orange directly into the chile crisp, then stir in the honey. (The chile crisp can be made 5 days ahead without the citrus zest; cover and chill. Bring to room temperature and stir in the zest just before serving.)
Cut the lemon and orange in halves and squeeze the juices into a medium bowl. Add the yogurt, season generously with salt and stir until smooth.
Spread the yogurt on a large platter, arrange the squash on top and drizzle with the chile crisp (or serve it on the side).
