Preheat oven to 375°F. Line a rimmed baking sheet with parchment paper.
Sprinkle each half of squash with 1/4 tsp. salt, then place facedown on prepared baking sheet along with onion quarters.
Combine turmeric, 1/2 tsp. salt, 1 tsp. allspice, and 1/8 tsp. cayenne in a large bowl. Add chicken and toss to coat. Arrange chicken skin side up alongside squash and onions on baking sheet. Brush tops of chicken with 1 Tbsp. olive oil. Bake until interior of squash and onions are fork-tender, chicken skin is crispy, and an instant read thermometer inserted into the thickest part of the chicken registers 165°F, 35–45 minutes; if chicken is done before squash, remove from oven and let rest until squash is finished cooking.
Meanwhile, whisk lemon juice, pomegranate molasses, and remaining 3 Tbsp. oil, 1/2 tsp. salt, 1/2 tsp. allspice, and 1/8 tsp. cayenne in a large bowl.
Finely chop onion quarters and transfer to bowl with dressing. Using a fork, scrape squash crosswise to pull strands from shells into bowl with dressing; discard shells. Add pistachios, pomegranate seeds, dill, and mint and toss to coat.
Divide chicken among plates, top with dill and mint, and serve alongside squash mixture.
Combine cucumber, yogurt, mint, and parsley in a medium bowl; season with hot sauce, salt, and pepper. Let sit at room temperature at least 15 minutes.
Do Ahead: Yogurt sauce can be made 1 hour ahead. Cover and chill.
